Theoretical analysis of long-lived radioactive waste in pressurized water reactor

Theoretical analysis of long-lived radioactive waste in pressurized water reactor 1 equal first-author 1 equal 1 https://orcid.org/0000-0002-9166-8625 1 1 https://orcid.org/0000-0002-8518-6443 1 corresp zhulong@mail.sysu.edu.cn zhulong@mail.sysu.edu.cn zhulong@mail.sysu.edu.cn https://orcid.org/0000-0002-5294-3823 Sino-French Institute of Nuclear Engineering and Technology, Sun Yat-sen University, Zhuhai 519082, China Sino-French Institute of Nuclear Engineering and Technology, Sun Yat-sen University, Zhuhai 519082, China Background: 2 The accelerator-driven subcritical transmutation system (ADS) is an advanced technology for the harmless disposal of nuclear waste. A theoretical analysis of the ingredients and content of nuclear waste, particularly long-lived waste in a pressurized water reactor (PWR), will provide important information for future spent fuel disposal. Purpose: 2 The present study is an attempt to investigate the yields of isotopes in the neutron-induced fission process and estimate the content of long-lived ingredients of nuclear waste in a PWR. Method: 2 We combined an approximation of the mass distribution of five Gaussians with the most probable charge model ( Z p model) to obtain the isotope yields in the 235 U(n,f) and 239 Pu(n,f) processes. The potential energy surface based on the concept of a di-nuclear system model was applied to an approximation using five Gaussian functions. A mathematical formula for the neutron spectrum in a PWR was established, and sets of differential equations were solved to calculate the content of long-lived nuclides in a PWR. Results: 2 The calculated isotopic fission yields were in good agreement with the experimental data. Except for 238 U, the contents of 239 Pu, 240 Pu, 241 Pu, 242 Pu, 237 Np, 235 U, and 236 U are predominant in the PWR after reaching a discharge burnup. In addition, some isotope pairs of heavy nuclei reach a similar value after stabilization, which can be explained by the decay chain and effective fission cross-sections. For fission fragments, we simulated the content evolution of some long-lived nuclides 90 Sr, 107 Pd, 135 Cs, and their isobars 90 Rb, 107 Rh, and 135 Xe during a fuel cycle in a PWR. The variations in the inventories of uranium and plutonium were in good agreement with the data in Daya Bay. Conclusion: 2 A new method is proposed for the prediction of the isotopic fission yield. The inventory of long-lived nuclides was analyzed and predicted after reaching a discharge burnup. Radiotoxicity PWR Five Gaussians Long-lived nuclides Fission fragments yields 1 Introduction 1 With the development of the nuclear industry, the amount of radioactive waste in storage worldwide has rapidly increased. An efficient prediction of the composition of spent fuel in nuclear reactors plays an important role in the design of facilities for radioactive waste management and has been a topic of significant interest. In [ 1 1 ], formulas and basic data were proposed for calculating the fission product radioactivity for a thermal neutron reactor. Similarly, codes for the fission products in the primary loop of a pressurized water reactor (PWR) were developed [ 2 2 , 3 3 ]. Simulations and calculations of the neutron flux densities have also been extensively studied [ 4 4 - 10 10 4-10 ]. In recent years, significant progress has been made in the mechanism of nuclear fission [ 11 11 - 18 18 11-18 ], which provides essential information for estimating the fission products in a nuclear reactor. The fission process can be described as a potential energy surface guiding the evolution of the nuclear shape [ 19 19 , 20 20 ]. According to the scission model, the shell effects can be reflected in the fission yield and can be distinguished according to different fission modes, which is related to the deformation of the nucleus at the scission point [ 20 20 , 21 21 ]. The improved scission-point model describes the charge distributions well [ 22 22 ]. The Gaussian fitting approach has been widely and successfully used in investigations into various fission products. By approximating different fission modes to Gaussian distributions, the dependence of the different fission modes on the mass and charge distributions can be studied. In general, the experimental data of a neutron-induced fission yield are measured with a fixed incident neutron energy within a certain energy range. Therefore, to study the continuous behavior of the fission yield with incident neutron energy, it is necessary to establish a continuous relationship between the yield and excitation energy. One of the most frequently adopted phenomenological approaches is to approximate the fission yield through a superposition of several Gaussian distributions [ 23 23 ]. However, most curve fitting results have been found through studies using a fixed neutron energy or specific compound nucleus. In this study, we theoretically analyze the evolution of fission products in a PWR using the multi-Gaussian function in combination with the most probable charge model and the concept of a di-nuclear system. The calculations are based on the parameters and information of a typical domestic PWR. The neutron spectrum plays an important role in studies on the fission product yields in a reactor. Wigeland et al. [ 24 24 ] divided the neutron spectrum into two energy ranges: thermal and fast, depending on the incident neutron energy. The epithermal energy range is also defined to better describe the spectrum [ 25 25 ]. The semi-empirical method can be useful for determination of the neutron spectrum [ 26 26 ]. Fig. 1 Fig. 1 shows the neutron spectrums in different nuclear reactors. It can be seen that a general incident neutron energy range in a reactor varies from 0.001 eV to 10 MeV. The formula used in this study was established to calculate the fission yields within this energy range. As mentioned above, the fission process in a nuclear reactor produces substantial fission fragments, which influence the reactivity of the reactor. However, fissile nuclides in a nuclear reactor continuously produce large amounts of long-lived nuclides, some of which, including 239 Pu, 241 Pu, and 233 U, can be extracted and reused in fission reactions [ 27 27 , 28 28 ]. However, some fission fragments, such as 90 Rb, 107 Rh, and 135 Xe are highly radioactive. The transmutation of minor actinides also contributes to long-term radiotoxicity [ 29 29 ]. The ADS system is flexible for lowering such waste [ 30 30 ], and the storage of spent nuclear fuel should be considered crucial [ 31 31 ]. Therefore, it is necessary to investigate the properties of the fission fragment evolution in a PWR in a theoretical manner. The remainder of this paper is organized as follows. In Sect. 2, the details of the theoretical method are described. The results and discussion are presented in Sect. 3. In Sec. 4, we provide some concluding remarks regarding this study. 2 Model 1 2.1 Multi-Gaussian functions 2 The Gaussian model was first proposed by Wahl[ 32 32 ], and this approach, based on the five Gaussians, has been widely used and continuously improved in estimations of fission fragment distributions. The expression for the five Gaussian superposition can be written as[ 33 33 ]: <math display="block" id="M1"><mrow><mi>y</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo><mo>=</mo><mstyle displaystyle="true"><munderover><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mn>5</mn></munderover></mstyle><mfrac><mrow><msub><mi>Y</mi><mi>i</mi></msub></mrow><mrow><msqrt><mrow><mn>2</mn><mi>π</mi></mrow></msqrt><msub><mi>σ</mi><mi>i</mi></msub></mrow></mfrac><mi>exp</mi><mrow><mo>{</mo><mrow><mo>−</mo><mfrac><mrow><msup><mrow><mo stretchy="false">(</mo><mi>A</mi><mo>−</mo><mfrac><mrow><msub><mi>A</mi><mtext>F</mtext></msub></mrow><mn>2</mn></mfrac><mo>+</mo><msub><mi>Δ</mi><mi>i</mi></msub><mo>+</mo><mi>n</mi><mi>t</mi><mo stretchy="false">)</mo></mrow><mn>2</mn></msup></mrow><mrow><mn>2</mn><msubsup><mi>σ</mi><mi>i</mi><mn>2</mn></msubsup></mrow></mfrac></mrow><mo>}</mo></mrow><mn>,</mn></mrow></math> where Y i represents the proportion of each Gaussian component. In addition, σ i and Δ i are the Gaussian parameters, 2 nt is considered as the total number of neutrons emitted during the fission process, and A F is the mass of the compound nucleus. Based on the symmetrical characteristic of the fission-fragment mass distribution with respect to <math id="M2"><mrow><mfrac><mrow><msub><mi>A</mi><mtext>F</mtext></msub></mrow><mn>2</mn></mfrac><mo>−</mo><mi>n</mi><mi>t</mi></mrow></math> , the relationship between the yields of light fragments and heavy fragments can be written as <math display="block" id="M3"><mrow><mi>y</mi><mo stretchy="false">(</mo><msub><mi>Z</mi><mtext>L</mtext></msub><mn>,</mn><msub><mi>A</mi><mtext>L</mtext></msub><mo stretchy="false">)</mo><mo>=</mo><mi>y</mi><mo stretchy="false">(</mo><msub><mi>Z</mi><mtext>F</mtext></msub><mo>−</mo><msub><mi>Z</mi><mtext>L</mtext></msub><mn>,</mn><msub><mi>A</mi><mtext>F</mtext></msub><mo>−</mo><mn>2</mn><mi>n</mi><mi>t</mi><mo>−</mo><msub><mi>A</mi><mtext>L</mtext></msub><mo stretchy="false">)</mo><mo>=</mo><mi>y</mi><mo stretchy="false">(</mo><msub><mi>Z</mi><mtext>H</mtext></msub><mn>,</mn><msub><mi>A</mi><mtext>H</mtext></msub><mo stretchy="false">)</mo><mn>,</mn></mrow></math> where Z L , A L , Z H , A H , Z F , and A F denote the proton number, mass number of light fragments, heavy fragments, and compound nucleus, respectively. It is considered that Y 1 = Y 5 , Y 2 = Y 4 , Δ 1 =-Δ 5 , Δ 2 =-Δ 4 , Δ 3 =0, σ 1 = σ 5 , and σ 2 = σ 4 . The proportions of these Gaussian components, Y i , are normalized such that <math id="M4"><mrow><mstyle displaystyle="true"><mo>∑</mo><mrow><msubsup><mrow/><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mn>5</mn></msubsup><msub><mi>Y</mi><mi>i</mi></msub></mrow></mstyle><mo>=</mo><mn>2</mn></mrow></math> . All of these Gaussian parameters are considered as functions of the excitation energy E * . 2.2 Potential energy surface 2 The parameters Y 2 and Y 3 in Eqs.(1) influence the variation tendency of the peak-to-valley ratio <math id="M5"><mrow><mfrac><mrow><msub><mi>y</mi><mrow><mtext>peak</mtext></mrow></msub></mrow><mrow><msub><mi>y</mi><mrow><mtext>valley</mtext></mrow></msub></mrow></mfrac></mrow></math> . In addition, y peak and y valley represent the peak and valley in the mass distribution for the product yields, respectively. Normally, the potential energy surface reflects the fission probability. Fig. 2 Fig. 2 shows the potential energy surface of 235 U in the di-nuclear system (DNS) concept, which is defined as follows[ 34 34 - 37 37 34-37 ] <math display="block" id="M6"><mrow><mtable columnalign="left"><mtr columnalign="left"><mtd columnalign="left"><mrow><mi>U</mi><mo stretchy="false">(</mo><msub><mi>Z</mi><mi>i</mi></msub><mn>,</mn><msub><mi>N</mi><mi>i</mi></msub><mn>,</mn><mi>R</mi><mo stretchy="false">)</mo></mrow></mtd><mtd columnalign="left"><mrow><mo>=</mo><msubsup><mi>U</mi><mtext>L</mtext><mrow><mtext>LD</mtext></mrow></msubsup><mo stretchy="false">(</mo><msub><mi>Z</mi><mtext>L</mtext></msub><mn>,</mn><msub><mi>N</mi><mtext>L</mtext></msub><mo stretchy="false">)</mo><mo>+</mo><msubsup><mi>U</mi><mtext>H</mtext><mrow><mtext>LD</mtext></mrow></msubsup><mo stretchy="false">(</mo><msub><mi>Z</mi><mtext>H</mtext></msub><mn>,</mn><msub><mi>N</mi><mtext>H</mtext></msub><mo stretchy="false">)</mo><mo>−</mo><msubsup><mi>U</mi><mrow><mtext>CN</mtext></mrow><mrow><mtext>LD</mtext></mrow></msubsup></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="left"><mrow/></mtd><mtd columnalign="left"><mrow><mo>+</mo><mi>δ</mi><msubsup><mi>U</mi><mtext>L</mtext><mrow><mtext>shell</mtext></mrow></msubsup><mo stretchy="false">(</mo><msub><mi>Z</mi><mtext>L</mtext></msub><mn>,</mn><msub><mi>N</mi><mtext>L</mtext></msub><mo stretchy="false">)</mo><mo>+</mo><mi>δ</mi><msubsup><mi>U</mi><mtext>H</mtext><mrow><mtext>shell</mtext></mrow></msubsup><mo stretchy="false">(</mo><msub><mi>Z</mi><mtext>H</mtext></msub><mn>,</mn><msub><mi>N</mi><mtext>H</mtext></msub><mo stretchy="false">)</mo></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="left"><mrow/></mtd><mtd columnalign="left"><mrow><mo>+</mo><msup><mi>V</mi><mtext>C</mtext></msup><mo stretchy="false">(</mo><msub><mi>Z</mi><mi>i</mi></msub><mn>,</mn><msub><mi>N</mi><mi>i</mi></msub><mn>,</mn><mi>R</mi><mo stretchy="false">)</mo><mo>+</mo><msup><mi>V</mi><mtext>N</mtext></msup><mo stretchy="false">(</mo><msub><mi>Z</mi><mi>i</mi></msub><mn>,</mn><msub><mi>N</mi><mi>i</mi></msub><mn>,</mn><mi>R</mi><mo stretchy="false">)</mo><mn>,</mn></mrow></mtd></mtr></mtable></mrow></math> where the indices i = L, H, and CN denote light, heavy, and compound nuclei, respectively. The potential energy is assumed to be the sum of the liquid drop ( <math id="M7"><mrow><msubsup><mi>U</mi><mi>i</mi><mrow><mtext>LD</mtext></mrow></msubsup></mrow></math> ) and microscopic shell correction ( <math id="M8"><mrow><mi>δ</mi><msubsup><mi>U</mi><mi>i</mi><mrow><mtext>shell</mtext></mrow></msubsup></mrow></math> ) for each DNS nucleus. In addition, the nuclear potential ( V N ) and Coulomb potential ( V C ) were used to describe the interaction between the fragments [ 34 34 ]. Relatively low potential energies result in high fission yields of the corresponding fragments[ 38 38 - 40 40 38-40 ]. Owing to shell closures of Z = 50 and N = 82, the valley with the minimum value V min can be seen, which results in relatively high fission yields of approximately Z = 50 and N = 82. By contrast, the potential energy V mid at the central position influences the fission yield at approximately Z = 46 and N = 72. We assume that <math id="M9"><mrow><mfrac><mrow><msub><mi>Y</mi><mn>2</mn></msub></mrow><mrow><msub><mi>Y</mi><mn>3</mn></msub></mrow></mfrac></mrow></math> is related to the values of V mid and V min , and can be written as <math display="block" id="M10"><mrow><mfrac><mrow><msub><mi>Y</mi><mn>2</mn></msub></mrow><mrow><msub><mi>Y</mi><mn>3</mn></msub></mrow></mfrac><mo>=</mo><mi>f</mi><mo stretchy="false">(</mo><msub><mi>V</mi><mrow><mtext>mid</mtext></mrow></msub><mo>−</mo><msub><mi>V</mi><mrow><mtext>min</mtext></mrow></msub><mo stretchy="false">)</mo><mn>.</mn></mrow></math> 2.3 Nuclear charge distribution 2 To obtain the isotopic fission yield, it is also assumed according to the most probable charge model[ 41 41 ] that the fission yield in the isobaric chains with mass number A follows a Gaussian dispersion: <math display="block" id="M11"><mrow><mi>y</mi><mo stretchy="false">(</mo><mi>Z</mi><mn>,</mn><mi>A</mi><mo stretchy="false">)</mo><mo>=</mo><mfrac><mrow><msub><mi>y</mi><mtext>A</mtext></msub></mrow><mrow><msqrt><mrow><mn>2</mn><mi>π</mi></mrow></msqrt><msub><mi>σ</mi><mtext>Z</mtext></msub></mrow></mfrac><mi>exp</mi><mrow><mo>(</mo><mrow><mfrac><mrow><mo>−</mo><msup><mrow><mo stretchy="false">(</mo><mi>Z</mi><mo>−</mo><msub><mi>Z</mi><mtext>p</mtext></msub><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><mn>2</mn></msup></mrow><mrow><mn>2</mn><msubsup><mi>σ</mi><mtext>Z</mtext><mn>2</mn></msubsup></mrow></mfrac></mrow><mo>)</mo></mrow><mn>,</mn></mrow></math> where y A and σ Z are Gaussian parameters. In addition, Z p ( A ) is the most probable charge in isobaric chains with mass number A . In[ 19 19 ], the fission yield of full isotopes for a different fission system is studied. It was found that the <math id="M12"><mrow><mfrac><mi>N</mi><mi>Z</mi></mfrac></mrow></math> ratio of heavy fragments with the most probable charge is closer to the <math id="M13"><mrow><mfrac><mi>N</mi><mi>Z</mi></mfrac></mrow></math> value of the compound nucleus. Therefore, we use a linear relation to describe the position of the most probable charge in the isobaric chains: <math display="block" id="M14"><mrow><msub><mi>Z</mi><mtext>p</mtext></msub><mo>=</mo><mi>k</mi><mo stretchy="false">(</mo><mi>A</mi><mo>−</mo><msub><mi>A</mi><mtext>H</mtext></msub><mo stretchy="false">)</mo><mo>+</mo><mi>b</mi><mn>.</mn></mrow></math> Considering the charge conservation and symmetry of the mass distribution, k and b are determined as follows: <math display="block" id="M15"><mrow><mi>k</mi><mo>=</mo><mo stretchy="false">(</mo><msub><mi>Z</mi><mtext>F</mtext></msub><mo>−</mo><mfrac><mrow><mn>2</mn><msub><mi>A</mi><mtext>H</mtext></msub><msub><mi>Z</mi><mtext>F</mtext></msub></mrow><mrow><msub><mi>A</mi><mtext>F</mtext></msub></mrow></mfrac><mo stretchy="false">)</mo><mo>/</mo><mo stretchy="false">(</mo><msub><mi>A</mi><mtext>L</mtext></msub><mo>−</mo><msub><mi>A</mi><mtext>H</mtext></msub><mo stretchy="false">)</mo><mn>,</mn></mrow></math> <math display="block" id="M16"><mrow><mi>b</mi><mo>=</mo><mfrac><mrow><msub><mi>A</mi><mtext>H</mtext></msub><msub><mi>Z</mi><mtext>F</mtext></msub></mrow><mrow><msub><mi>A</mi><mtext>F</mtext></msub></mrow></mfrac><mn>,</mn></mrow></math> where A L and A H represent the mass numbers of light and heavy fragments, respectively. In an isobaric chain, <math display="block" id="M17"><mrow><mstyle displaystyle="true"><munder><mo>∑</mo><mi>i</mi></munder></mstyle><mi>y</mi><mrow><mo>(</mo><mrow><msub><mi>Z</mi><mi>i</mi></msub><mn>,</mn><mi>A</mi></mrow><mo>)</mo></mrow><mo>=</mo><mi>y</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo><mn>,</mn></mrow></math> where y ( A ) is the mass distribution and y ( Z i , A ) is the yield of the fission fragment with proton number Z i in the isobaric chain with mass number A . The determination of parameter σ Z involves the study of 〈 σ Z 〉 and <math id="M18"><mrow><mo>〈</mo><msubsup><mi>σ</mi><mtext>Z</mtext><mn>2</mn></msubsup><mo>〉</mo></mrow></math> for different fission systems. To make the formula more universal, σ Z in Eq.(5) is replaced by the average value (〈 σ Z 〉) for all isobaric chains, which results in the absence of odd-even effects. In addition, following the description in [ 42 42 ], for a different fission system, <math id="M19"><mrow><mo>〈</mo><msubsup><mi>σ</mi><mtext>Z</mtext><mn>2</mn></msubsup><mo>〉</mo></mrow></math> is energy independent. In addition, for a compound nucleus with the same proton number, 〈 σ Z 〉 can be assumed to be constant. In this study, 〈 σ Z 〉 = 0.55[ 42 42 ]. 2.4 Prediction of components in spent fuel 2 The inventory of components in a PWR can be obtained through the following formula[ 43 43 , 44 44 ]: <math display="block" id="M20"><mrow><mtable columnalign="left"><mtr columnalign="left"><mtd columnalign="left"><mrow><mfrac><mrow><mtext>d</mtext><msub><mi>N</mi><mi>i</mi></msub><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></mrow><mrow><mtext>d</mtext><mi>t</mi></mrow></mfrac><mo>=</mo><mstyle displaystyle="true"><munder><mo>∑</mo><mi>l</mi></munder></mstyle><msub><mi>N</mi><mi>l</mi></msub><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>⋅</mo><msub><mi>σ</mi><mrow><mtext>f</mtext><mn>,</mn><mi>l</mi></mrow></msub><mo>⋅</mo><mi>Φ</mi><mo>⋅</mo><msub><mi>y</mi><mrow><mi>i</mi><mn>,</mn><mi>l</mi></mrow></msub><mo>+</mo><mstyle displaystyle="true"><munder><mo>∑</mo><mi>k</mi></munder></mstyle><msub><mi>λ</mi><mi>k</mi></msub><mo>⋅</mo><msub><mi>K</mi><mrow><mi>p</mi><mn>,</mn><mi>k</mi></mrow></msub><mo>⋅</mo><msub><mi>N</mi><mi>k</mi></msub><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>+</mo></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="left"><mrow><mstyle displaystyle="true"><munder><mo>∑</mo><mi>j</mi></munder></mstyle><msub><mi>σ</mi><mrow><mi>c</mi><mn>,</mn><mi>j</mi></mrow></msub><mo>⋅</mo><mi>Φ</mi><mo>⋅</mo><msub><mi>N</mi><mi>j</mi></msub><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>−</mo><mrow><mo>(</mo><mrow><msub><mi>λ</mi><mi>i</mi></msub><mo>+</mo><msub><mi>σ</mi><mrow><mi>a</mi><mn>,</mn><mi>i</mi></mrow></msub><mo>⋅</mo><mi>Φ</mi></mrow><mo>)</mo></mrow><mo>⋅</mo><msub><mi>N</mi><mi>i</mi></msub><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mn>,</mn></mrow></mtd></mtr></mtable></mrow></math> where N i is the nuclide number density of isotope i . In addition, σ f, l , σ c , j , and σ a , i represent the microscopic effective fission cross section, gamma capture cross section, and absorption cross section, respectively. Moreover, Φ denotes the neutron flux in the reactor, y i , l is the yield of fragment i produced by the fissile nuclide l , λ i is the decay constant of nuclide i , and K p , k is the decay branching ratio of parent nuclide k . It can be seen that the production process of nuclide i contains several parts. For heavy nuclides, the formation from the decay process and neutron capture are considered. For fission fragments, the fission yield was also considered. The consumption process includes decay and neutron absorption. The effective microscopic reaction cross section is defined as[ 43 43 ] <math display="block" id="M21"><mrow><msub><mi>σ</mi><mrow><mi>x</mi><mn>,</mn><mi>r</mi></mrow></msub><mo>=</mo><mstyle displaystyle="true"><mrow><msubsup><mo>∫</mo><mrow><mn>0.001</mn><mtext>eV</mtext></mrow><mrow><mn>10</mn><mtext>MeV</mtext></mrow></msubsup></mrow></mstyle><mtext>d</mtext><msub><mi>E</mi><mtext>n</mtext></msub><mi/><msub><mi>σ</mi><mrow><mi>x</mi><mn>,</mn><mi>r</mi></mrow></msub><mo stretchy="false">(</mo><msub><mi>E</mi><mtext>n</mtext></msub><mo stretchy="false">)</mo><mi>χ</mi><mo stretchy="false">(</mo><msub><mi>E</mi><mtext>n</mtext></msub><mo stretchy="false">)</mo><mn>.</mn></mrow></math> In the above equation, σ x, r is defined as the effective microscopic cross section for the reaction r of nuclide x . In addition, E n is the neutron energy in the reactor, and χ is the neutron distribution probability density, which is simplified as space-independent. The integral bound at 10 MeV and 0.001 eV is determined based on the energy range of the neutron spectrum in the PWR. We can obtain the following equation by a change in variable u =ln ( E 0 / E n ), which provides a new integration boundary, which is approximated as 23.03 0. <math display="block" id="M22"><mrow><msub><mi>σ</mi><mrow><mi>x</mi><mn>,</mn><mi>r</mi></mrow></msub><mo>=</mo><mstyle displaystyle="true"><mrow><msubsup><mo>∫</mo><mn>0</mn><mrow><mn>23.03</mn></mrow></msubsup></mrow></mstyle><mtext>d</mtext><mi>u</mi><mi/><msub><mi>σ</mi><mrow><mtext>x</mtext><mn>,</mn><mtext>r</mtext></mrow></msub><mo stretchy="false">(</mo><mi>u</mi><mo stretchy="false">)</mo><mo stretchy="false">[</mo><msub><mi>E</mi><mtext>n</mtext></msub><mi>χ</mi><mo stretchy="false">(</mo><msub><mi>E</mi><mtext>n</mtext></msub><mo stretchy="false">)</mo><mo stretchy="false">]</mo><mo stretchy="false">(</mo><mi>u</mi><mo stretchy="false">)</mo><mn>,</mn></mrow></math> where E 0 = 10 MeV, u is often referred to as lethargy, and ϕ = E n χ( E n ) is referred to as the normalized neutron spectrum. 3 Results and discussion 1 3.1 Determination of Gaussian parameters 2 The fitting results of 233,235,238 U(n,f) and 239,240,241 Pu(n,f) are shown with experimental data from [ 45 45 - 55 55 45-55 ] in Figs. 3 Figs. 3 and 4 4 . We applied the curve fitting results using the least squares method and obtained the parameters of the Gaussian functions in Eq.(1), which can be used in the study of neutron-induced fission within the excitation energy range of the thermal energy of up to 10 MeV. The parameter Y 3 increases exponentially with increasing excitation energy, which results in an increase in the peak-to-valley ratio directly. Under the premise that <math id="M23"><mrow><mfrac><mrow><msub><mi>A</mi><mtext>F</mtext></msub></mrow><mn>2</mn></mfrac><mo>−</mo><mi>n</mi><mi>t</mi></mrow></math> is the symmetric axis in the mass distribution, Δ 3 is fixed at 0. Δ i and σ i are assumed to be constant for a fixed fission system. In addition, 2 nt denotes the total number of neutrons emitted, which increases slightly with an increase in the excitation energy. For uranium, Δ i and σ i exhibit a linear relationship with the mass number of a compound nuclei. The Gaussian parameters are expressed as a function of the mass number and excitation energy. The parameters in Eq.(1) can be expressed as follows: <math display="block" id="M24"><mrow><mrow><mo>{</mo><mrow><mtable><mtr><mtd><mrow><msub><mi>σ</mi><mi>i</mi></msub><mo>=</mo><mi>p</mi><mi>a</mi><mi>r</mi><mn>1</mn><mo stretchy="false">(</mo><msub><mi>A</mi><mtext>F</mtext></msub><mo>−</mo><mn>1</mn><mo stretchy="false">)</mo><mo>+</mo><mi>p</mi><mi>a</mi><mi>r</mi><mn>2,</mn></mrow></mtd></mtr><mtr><mtd><mrow/></mtd></mtr><mtr><mtd><mrow><msub><mi>Δ</mi><mi>i</mi></msub><mo>=</mo><mi>p</mi><mi>a</mi><mi>r</mi><mn>1</mn><mo stretchy="false">(</mo><msub><mi>A</mi><mtext>F</mtext></msub><mo>−</mo><mn>1</mn><mo stretchy="false">)</mo><mo>+</mo><mi>p</mi><mi>a</mi><mi>r</mi><mn>2,</mn></mrow></mtd></mtr><mtr><mtd><mrow/></mtd></mtr><mtr><mtd><mrow><mi>n</mi><mi>t</mi><mo>=</mo><mrow><mo>(</mo><mrow><mi>p</mi><mi>a</mi><mi>r</mi><mn>1</mn><mo stretchy="false">(</mo><msub><mi>A</mi><mtext>F</mtext></msub><mo>−</mo><mn>1</mn><mo stretchy="false">)</mo><mo>+</mo><mi>p</mi><mi>a</mi><mi>r</mi><mn>2</mn></mrow><mo>)</mo></mrow><msup><mi>E</mi><mo>*</mo></msup><mo>+</mo><mi>p</mi><mi>a</mi><mi>r</mi><mn>3,</mn></mrow></mtd></mtr><mtr><mtd><mrow/></mtd></mtr><mtr><mtd><mrow><msub><mi>Y</mi><mn>3</mn></msub><mo>=</mo><mi>exp</mi><mo>{</mo><mo stretchy="false">(</mo><mi>p</mi><mi>a</mi><mi>r</mi><mn>1</mn><mo stretchy="false">(</mo><msub><mi>A</mi><mtext>F</mtext></msub><mo>−</mo><mn>1</mn><mo stretchy="false">)</mo><mo>+</mo><mi>p</mi><mi>a</mi><mi>r</mi><mn>2</mn><mo stretchy="false">)</mo><msup><mi>E</mi><mo>*</mo></msup></mrow></mtd></mtr><mtr><mtd><mrow><mo>+</mo><mo stretchy="false">(</mo><mi>p</mi><mi>a</mi><mi>r</mi><mn>3</mn><mo stretchy="false">(</mo><msub><mi>A</mi><mtext>F</mtext></msub><mo>−</mo><mn>1</mn><mo stretchy="false">)</mo><mo>+</mo><mi>p</mi><mi>a</mi><mi>r</mi><mn>4</mn><mo stretchy="false">)</mo><mo>}</mo><mn>,</mn></mrow></mtd></mtr><mtr><mtd><mrow/></mtd></mtr><mtr><mtd><mrow><msub><mi>Y</mi><mn>2</mn></msub><mo>=</mo><msub><mi>Y</mi><mn>3</mn></msub><msup><mrow><mrow><mo>{</mo><mrow><mo stretchy="false">(</mo><msub><mi>V</mi><mrow><mtext>mid</mtext></mrow></msub><mo>−</mo><msub><mi>V</mi><mrow><mtext>min</mtext></mrow></msub><mo stretchy="false">)</mo><mo>/</mo><mo stretchy="false">(</mo><mn>0.055</mn><mo stretchy="false">(</mo><msub><mi>A</mi><mtext>F</mtext></msub><mo>−</mo><mn>1</mn><mo stretchy="false">)</mo><mo>−</mo><mn>9.215</mn><mo stretchy="false">)</mo></mrow><mo>}</mo></mrow></mrow><mrow><mn>5.7</mn></mrow></msup><mn>,</mn></mrow></mtd></mtr><mtr><mtd><mrow/></mtd></mtr><mtr><mtd><mrow><msub><mi>Y</mi><mn>1</mn></msub><mo>=</mo><mn>1</mn><mo>−</mo><msub><mi>Y</mi><mn>2</mn></msub><mo>−</mo><mfrac><mrow><msub><mi>Y</mi><mn>3</mn></msub></mrow><mn>2</mn></mfrac><mn>,</mn></mrow></mtd></mtr><mtr><mtd><mrow/></mtd></mtr><mtr><mtd><mrow/></mtd></mtr></mtable></mrow></mrow></mrow></math> where par 1, par 2, par 3, and par 4 are listed in Table 1 Table 1 . In addition, V mid and V min denote the values of potential energy with a symmetry configuration and combinations with proton and neutron shell closures, respectively, as shown in Fig.2 Fig.2 . Moreover, E * is the excitation energy of the fission system. 1 1 parameters 1 1 par1 1 1 par2 1 1 par3 1 1 par4 1 1 σ 1 1 1 0.421 ± 0.017 1 1 -94.818 ± 4.039 1 1 σ 2 1 1 -0.482 ± 0.041 1 1 117.18 ± 9.57 1 1 σ 3 1 1 0 1 1 20 1 1 Δ 1 1 1 -0.866 ± 0.050 1 1 229.12 ± 11.80 1 1 Δ 2 1 1 -0.895 ± 0.046 1 1 228.89 ± 10.73 1 1 nt 1 1 -0.0033 ± 0.0002 1 1 0.837 ± 0.043 1 1 0.957 ± 0.004 1 1 Y 3 ( A F ≤236) 1 1 -0.00805 ± 0.00001 1 1 2.155 ± 0.002 1 1 -0.1106 ± 0.0001 1 1 20.05 ± 0.03 1 1 Y 3 ( A F > 236) 1 1 -0.00250 ± 0.00001 1 1 0.851 ± 0.002 1 1 -0.0146 ± 0.0003 1 1 -2.50 ± 0.05 For the case of plutonium isotopes, because the mass numbers of 239 Pu, 240 Pu, and 241 Pu are close to each other, the dependence of the Gaussian parameters on the mass number is not obvious. An assumption of the mass number correlation is not made for the Gaussian parameters. In addition, <math id="M25"><mrow><mfrac><mrow><msub><mi>Y</mi><mn>2</mn></msub></mrow><mrow><msub><mi>Y</mi><mn>3</mn></msub></mrow></mfrac></mrow></math> can be obtained using the following expression: <math display="block" id="M26"><mrow><mfrac><mrow><msub><mi>Y</mi><mn>2</mn></msub></mrow><mrow><msub><mi>Y</mi><mn>3</mn></msub></mrow></mfrac><mtext>=</mtext><msup><mrow><mrow><mo>(</mo><mrow><mfrac><mrow><msub><mi>V</mi><mrow><mtext>mid</mtext></mrow></msub><mo>−</mo><msub><mi>V</mi><mrow><mtext>min</mtext></mrow></msub></mrow><mrow><mn>2.9</mn></mrow></mfrac></mrow><mo>)</mo></mrow></mrow><mrow><mn>7.5</mn></mrow></msup><mn>.</mn></mrow></math> The value of Y i depends on the excitation energy. The parameters σ i and Δ i were chosen to be fixed by changing the excitation energy. The values of the Gaussian parameters for 239 Pu, 240 Pu, and 241 Pu are listed in Table 2 Table 2 . 1 1 Nuclide 1 1 σ 1 1 1 σ 2 1 1 σ 3 1 1 Δ 1 1 1 Δ 2 1 1 239 Pu 1 1 4 1 1 3.5 1 1 17.5 1 1 23.5 1 1 16 1 1 240 Pu 1 1 4 1 1 3.5 1 1 17.5 1 1 23.5 1 1 16 1 1 241 Pu 1 1 3.5 1 1 3.2 1 1 17.5 1 1 22.9 1 1 14.9 3.2 Comparisons with experimental data 2 Owing to the scarce independent yield data of the fission products, we fit the data to the cumulative yields to investigate the energy dependence. In Figs. 5 Figs. 5 and 6 6 , the calculated isotope yields from the fitted mass distributions of 235 U(n th ,f) and 239 Pu(n th ,f) were compared with the experimental data[ 56 56 - 59 59 56-59 ]. Numerical comparisons of the experimental independent and simulated yields for 235 U(n th ,f) and 239 Pu(n th ,f) are shown in Tables 3 Tables 3 and 4 4 . As shown in Tables 3 Tables 3 and 4 4 , we present the errors in the experimental fragments and compare the simulation results with the experimental data. Within the permissible range of errors, the thermal neutron-induced fission yields predicted by our simulations are in good agreement with the experimental data, which proves the validity of our methods and models. 1 1 Nuclide 1 1 Experimental yield 1 1 Simulated yield 1 1 Relative difference 1 1 83 Br 1 1 0.00020 ±0.00004 1 1 0.00016 1 1 -0.210 1 1 84 Br 1 1 0.00048 ±0.00007 1 1 0.00111 1 1 1.313 1 1 85 Br 1 1 0.00224 ±0.00009 1 1 0.00480 1 1 1.143 1 1 86 Br 1 1 0.00674 ±0.00024 1 1 0.01237 1 1 0.835 1 1 88 Br 1 1 0.01428 ±0.00026 1 1 0.01911 1 1 0.338 1 1 89 Br 1 1 0.01512 ±0.00052 1 1 0.01814 1 1 0.200 1 1 90 Br 1 1 0.01067 ±0.00044 1 1 0.01029 1 1 -0.036 1 1 91 Br 1 1 0.00287 ±0.00021 1 1 0.00359 1 1 0.251 1 1 85 Kr 1 1 0.00039 ±0.00007 1 1 0.00010 1 1 -0.744 1 1 86 Kr 1 1 0.00113 ±0.00009 1 1 0.00087 1 1 -0.228 1 1 87 Kr 1 1 0.00468 ±0.00024 1 1 0.00456 1 1 -0.026 1 1 88 Kr 1 1 0.01769 ±0.00054 1 1 0.01465 1 1 -0.172 1 1 89 Kr 1 1 0.03489 ±0.00042 1 1 0.02814 1 1 -0.193 1 1 90 Kr 1 1 0.04782 ±0.00070 1 1 0.03320 1 1 -0.306 1 1 91 Kr 1 1 0.03384 ±0.00107 1 1 0.02427 1 1 -0.283 1 1 92 Kr 1 1 0.01644 ±0.00077 1 1 0.01082 1 1 -0.342 1 1 93 Kr 1 1 0.00402 ±0.00022 1 1 0.00305 1 1 -0.241 1 1 94 Kr 1 1 0.00092 ±0.00019 1 1 0.00054 1 1 -0.414 1 1 87 Rb 1 1 0.00041 ±0.00012 1 1 0.00004 1 1 -0.903 1 1 88 Rb 1 1 0.00069 ±0.00014 1 1 0.00043 1 1 -0.371 1 1 89 Rb 1 1 0.00252 ±0.00034 1 1 0.00282 1 1 0.119 1 1 90 Rb 1 1 0.00860 ±0.00065 1 1 0.01127 1 1 0.310 1 1 91 Rb 1 1 0.02347 ±0.00092 1 1 0.02789 1 1 0.188 1 1 92 Rb 1 1 0.03354 ±0.00045 1 1 0.04208 1 1 0.255 1 1 93 Rb 1 1 0.03167 ±0.00079 1 1 0.04021 1 1 0.270 1 1 94 Rb 1 1 0.01557 ±0.00051 1 1 0.02402 1 1 0.543 1 1 95 Rb 1 1 0.00636 ±0.00024 1 1 0.00893 1 1 0.404 1 1 96 Rb 1 1 0.00115 ±0.00020 1 1 0.00211 1 1 0.835 1 1 97 Rb 1 1 0.00037 ±0.00018 1 1 0.00031 1 1 -0.170 1 1 90 Sr 1 1 0.00081 ±0.00024 1 1 0.00014 1 1 -0.827 1 1 91 Sr 1 1 0.00229 ±0.00027 1 1 0.00118 1 1 -0.485 1 1 92 Sr 1 1 0.00959 ±0.00067 1 1 0.00600 1 1 -0.374 1 1 93 Sr 1 1 0.02571 ±0.00084 1 1 0.01942 1 1 -0.245 1 1 94 Sr 1 1 0.04506 ±0.00052 1 1 0.03928 1 1 -0.128 1 1 95 Sr 1 1 0.04756 ±0.00056 1 1 0.04940 1 1 0.039 1 1 96 Sr 1 1 0.03701 ±0.00075 1 1 0.03958 1 1 0.069 1 1 97 Sr 1 1 0.01794 ±0.00074 1 1 0.01949 1 1 0.086 1 1 98 Sr 1 1 0.00818 ±0.00046 1 1 0.00593 1 1 -0.275 1 1 99 Sr 1 1 0.00155 ±0.00022 1 1 0.00112 1 1 -0.277 1 1 138 Xe 1 1 0.0473 ±0.0032 1 1 0.04865 1 1 0.029 1 1 139 Xe 1 1 0.048 ±0.003 1 1 0.04381 1 1 -0.085 1 1 140 Xe 1 1 0.0355 ±0.0013 1 1 0.02477 1 1 -0.302 1 1 141 Xe 1 1 0.01360 ±0.00060 1 1 0.00868 1 1 -0.362 1 1 142 Xe 1 1 0.00447 ±0.00020 1 1 0.00194 1 1 -0.566 1 1 137 Cs 1 1 0.00061 ±0.00006 1 1 0.00130 1 1 1.145 1 1 138 Cs 1 1 0.00532 ±0.00021 1 1 0.00627 1 1 0.179 1 1 139 Cs 1 1 0.01430 ±0.00055 1 1 0.01912 1 1 0.337 1 1 140 Cs 1 1 0.0284 ±0.0012 1 1 0.03661 1 1 0.289 1 1 141 Cs 1 1 0.031 ±0.001 1 1 0.04345 1 1 0.397 1 1 142 Cs 1 1 0.02684 ±0.00072 1 1 0.03280 1 1 0.222 1 1 143 Cs 1 1 0.01355 ±0.00037 1 1 0.01521 1 1 0.123 1 1 144 Cs 1 1 0.00428 ±0.00010 1 1 0.00434 1 1 0.014 1 1 145 Cs 1 1 0.00085 ±0.00003 1 1 0.00077 1 1 -0.101 1 1 146 Cs 1 1 0.000076 ±0.000006 1 1 0.00009 1 1 0.070 1 1 Nuclide 1 1 Experimental yield 1 1 Simulated yield 1 1 Relative difference 1 1 89 Rb 1 1 0.00271 ±0.00088 1 1 0.00249 1 1 -0.081 1 1 91 Rb 1 1 0.0136 ±0.0021 1 1 0.01834 1 1 0.349 1 1 92 Rb 1 1 0.0164 ±0.0028 1 1 0.02279 1 1 0.390 1 1 93 Rb 1 1 0.0195 ±0.0030 1 1 0.01707 1 1 -0.125 1 1 94 Rb 1 1 0.0097 ±0.0020 1 1 0.00759 1 1 -0.218 1 1 95 Rb 1 1 0.0075 ±0.0018 1 1 0.00209 1 1 -0.721 1 1 97 Rb 1 1 0.00057 ±0.00089 1 1 0.000038 1 1 -0.934 1 1 93 Sr 1 1 0.02310 ±0.00025 1 1 0.02076 1 1 -0.101 1 1 94 Sr 1 1 0.03168 ±0.00034 1 1 0.0318 1 1 -0.004 1 1 95 Sr 1 1 0.02820 ±0.00031 1 1 0.03024 1 1 0.072 1 1 96 Sr 1 1 0.01927 ±0.00021 1 1 0.01771 1 1 -0.081 1 1 98 Sr 1 1 0.00331 ±0.00013 1 1 0.00152 1 1 -0.541 1 1 99 Sr 1 1 0.000400 ±0.000004 1 1 0.00022 1 1 -0.445 1 1 101 Tc 1 1 0.00014 ±0.00009 1 1 0.000003 1 1 -0.980 1 1 103 Tc 1 1 0.00241 ±0.00097 1 1 0.00043 1 1 -0.821 1 1 104 Tc 1 1 0.00536 ±0.00083 1 1 0.00247 1 1 -0.539 1 1 105 Tc 1 1 0.01086 ±0.00251 1 1 0.00818 1 1 -0.247 1 1 104 Ru 1 1 0.00004 ±0.00001 1 1 0.000006 1 1 -0.852 1 1 105 Ru 1 1 0.00008 ±0.00001 1 1 0.00007 1 1 -0.154 1 1 106 Ru 1 1 0.00007 ±0.00003 1 1 0.00046 1 1 -0.363 1 1 107 Ru 1 1 0.00253 ±0.00028 1 1 0.00179 1 1 -0.292 1 1 108 Ru 1 1 0.00619 ±0.00025 1 1 0.00409 1 1 -0.339 1 1 109 Ru 1 1 0.00674 ±0.00046 1 1 0.00549 1 1 -0.185 1 1 142 La 1 1 0.00406 ±0.00006 1 1 0.00447 1 1 0.101 1 1 143 La 1 1 0.01045 ±0.00016 1 1 0.01253 1 1 0.199 1 1 144 La 1 1 0.01228 ±0.00024 1 1 0.02129 1 1 0.734 1 1 145 La 1 1 0.01796 ±0.00027 1 1 0.02148 1 1 0.196 1 1 146 La 1 1 0.00742 ±0.00010 1 1 0.01315 1 1 0.772 1 1 147 La 1 1 0.00655 ±0.00011 1 1 0.00472 1 1 -0.279 1 1 148 La 1 1 0.00191 ±0.00003 1 1 0.00101 1 1 -0.471 1 1 145 Ce 1 1 0.00430 ±0.00008 1 1 0.00437 1 1 0.016 1 1 146 Ce 1 1 0.00793 ±0.00019 1 1 0.00922 1 1 0.163 1 1 147 Ce 1 1 0.01619 ±0.00031 1 1 0.01141 1 1 -0.295 1 1 148 Ce 1 1 0.00920 ±0.00022 1 1 0.00838 1 1 -0.089 To further verify the feasibility of our method in the establishment of fission product charge distribution, we compared the calculated charge distribution and experimental data in the reactions 235 U(n th ,f) and 239 Pu(n th ,f), as shown in Fig. 7 Fig. 7 ). The experimental data were obtained from [ 56 56 ]. The calculated peak is slightly higher than the experimental data. However, both the calculated yield and experimental data reached the maximum at approximately Z = 54, which might suggest a transition of shell closure from Z = 50 to Z = 54. . 3.3 Neutron spectrum 2 Based on the Daya Bay nuclear power plant, a neutron spectrum was established by modeling an actual reactor core using MCNP4C[ 60 60 ]. The neutron spectrum in the nuclear reactor generally ranges from 0.001 eV to 10 MeV and is separated into three parts: thermal neutron energy, epithermal (intermediate) neutron energy, and fast neutron energy. Thermal neutron energy ranges from 0.001 to 0.1 eV, where most fission reactions take place in a PWR. The fast neutron energy ranges from 10 5 eV to 10 MeV. Most of these neutrons are emitted before and after fission. The energy of epithermal neutrons is between 0.1 and 10 5 eV. In thermal neutron reactors, neutrons emitted during the fission process have an average energy of 2 MeV. The neutrons lose energy by elastic or inelastic collisions with nuclei in the moderator medium until they become thermal neutrons. Most fission reactions in thermal nuclear reactors are induced through thermal neutrons. The neutron spectrum is often referred to as <math display="block" id="M27"><mrow><mi>ϕ</mi><mo>=</mo><msub><mi>E</mi><mtext>n</mtext></msub><mo>×</mo><mi>χ</mi><mo stretchy="false">(</mo><msub><mi>E</mi><mtext>n</mtext></msub><mo stretchy="false">)</mo><mn>,</mn></mrow></math> where ϕ is the neutron spectrum, χ ( E n ) is the neutron distribution probability density at an energy of E n . A piecewise function was used to describe the neutron spectrum. In the thermal neutron region, χ ( E n ) is often approximated using a Maxwellian-Boltzmann distribution. A method of superposing five to seven partial Maxwellian distributions to represent the neutron spectrum within the thermal and epithermal range was proposed in [ 25 25 ]. We fo und that a single Maxwellian distribution is sufficient to describe the neutron spectrum within the thermal range. The multi-Maxwellian distribution mainly takes effect in the transition part between the thermal and epithermal ranges. The formula used is as follows: <math display="block" id="M28"><mrow><mi>χ</mi><mo stretchy="false">(</mo><msub><mi>E</mi><mtext>n</mtext></msub><mo stretchy="false">)</mo><mo>=</mo><msubsup><mi>α</mi><mtext>M</mtext><mn>2</mn></msubsup><mi>exp</mi><mo stretchy="false">(</mo><mo>−</mo><msub><mi>E</mi><mtext>n</mtext></msub><msub><mi>α</mi><mtext>M</mtext></msub><mo stretchy="false">)</mo><msub><mi>E</mi><mtext>n</mtext></msub><mn>,</mn></mrow></math> where α M is related to the temperature of the moderating medium [ 25 25 ]: <math display="block" id="M29"><mrow><msub><mi>α</mi><mtext>M</mtext></msub><mo>=</mo><mfrac><mn>1</mn><mrow><msub><mi>T</mi><mtext>m</mtext></msub></mrow></mfrac><mn>.</mn></mrow></math> Here, T m is the temperature of the moderating medium. The experimental thermal neutron energy distribution in the case of a water-moderated reactor was compared in [ 67 67 ]. Fig. 8 Fig. 8 shows a comparison of the experimental thermal neutron spectrum in water moderated reactors at 291.15 and 371.15 K with the calculation results in this study. It can be seen that the calculated results can reproduce the experimental data quite well. Within the epithermal neutron energy range, the probability density of the neutron energy distribution follows the <math id="M30"><mrow><mfrac><mn>1</mn><mrow><msub><mi>E</mi><mtext>n</mtext></msub></mrow></mfrac></mrow></math> law. To make the transition smoother between the three different regions and more suitable to the results in [ 60 60 ], the probability density in this region is assumed as <math display="block" id="M31"><mrow><mi>χ</mi><mo stretchy="false">(</mo><msub><mi>E</mi><mtext>n</mtext></msub><mo stretchy="false">)</mo><mo>=</mo><mi>C</mi><mfrac><mn>1</mn><mrow><msup><mrow><mo stretchy="false">(</mo><msub><mi>E</mi><mtext>n</mtext></msub><mo stretchy="false">)</mo></mrow><mrow><mn>0.9</mn></mrow></msup></mrow></mfrac><mo>+</mo><mfrac><mrow><mn>0.02</mn></mrow><mrow><msup><mrow><mo stretchy="false">(</mo><msub><mi>E</mi><mtext>n</mtext></msub><mo stretchy="false">)</mo></mrow><mrow><mn>2.1</mn></mrow></msup></mrow></mfrac><mo>+</mo><mfrac><mn>7</mn><mrow><msup><mrow><mn>10</mn></mrow><mn>7</mn></msup><mo>−</mo><msub><mi>E</mi><mtext>n</mtext></msub></mrow></mfrac><mn>,</mn></mrow></math> where C is a constant determined by the continuous condition at the boundary of the thermal and epithermal energy ranges. In the fast neutron energy range of the PWR, χ ( E n ) is approximated by the thermal neutron-induced fission spectrum of 235 U. The experimental data and curve fitting results of the 235 U fission spectrum are presented in [ 61 61 , 62 62 ]. In [ 62 62 ], the prompt neutron spectrum from thermal neutron-induced fission in 235 U using the recoil proton method was recently measured. In [ 61 61 ], a photographic plate method and time-of-flight method are employed. Two different formulas are used in curve fitting, all of which fit well with the experimental data. In this study, the fission spectrum of 235 U is considered as a linear combination of the two formulas and can be written as <math display="block" id="M32"><mrow><mtable columnalign="left"><mtr columnalign="left"><mtd columnalign="left"><mrow><mi>χ</mi><mo stretchy="false">(</mo><msub><mi>E</mi><mtext>n</mtext></msub><mo stretchy="false">)</mo><mo>=</mo><msub><mi>C</mi><mn>1</mn></msub><mo stretchy="false">(</mo><mi>ω</mi><msqrt><mrow><msub><mi>E</mi><mtext>n</mtext></msub></mrow></msqrt><msup><mi>e</mi><mrow><mo>−</mo><msub><mi>E</mi><mtext>n</mtext></msub><mo>/</mo><msub><mi>E</mi><mtext>M</mtext></msub></mrow></msup></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="left"><mrow><mo>+</mo><mrow><mo>(</mo><mrow><mn>1</mn><mo>−</mo><mi>ω</mi></mrow><mo>)</mo></mrow><msup><mi>e</mi><mrow><mo>−</mo><msub><mi>E</mi><mtext>n</mtext></msub><mo>/</mo><mi>a</mi></mrow></msup><mi>sinh</mi><msqrt><mrow><mi>b</mi><msub><mi>E</mi><mtext>n</mtext></msub></mrow></msqrt><mo stretchy="false">)</mo><mn>,</mn></mrow></mtd></mtr></mtable></mrow></math> where ω denotes the weight, and E M , a , and b are constants. These were all determined in [ 63 63 ] using the least-squares method. In addition, C 1 is determined based on the continuous condition at the boundary of the epithermal and fast energy ranges. The normalized neutron spectrum within all energy ranges was multiplied by a normalized constant C norm . In this study, the boundaries between the three energy ranges were 10 -0.6 and 10 5.7 eV. In addition, T m is taken as 563.15 K. In Fig. 9 Fig. 9 , the normalized neutron spectrum in the PWR was compared with the calculated results obtained in [ 60 60 ]. In our mathematical description of the neutron spectrum, the resonance in the epithermal energy range was not considered. 3.4 Effective cross section in PWR 2 In Table 5 Table 5 , we list the effective fission and neutron capture cross-sections of several heavy nuclides calculated using Eq.(11). The cross-section data are taken from the ENDF library, and the neutron spectrum proposed above (Eqs. (16), (18), and (19)) are used to calculate the effective cross sections. 1 1 Nuclide 1 1 σ f (barns) 1 1 σ c (barns) 1 1 σ f / σ c 1 1 σ f / σ c in Ref.[ 43 43 ] 1 1 242 Cm 1 1 1.2005 1 1 3.0541 1 1 0.3931 1 1 0.2653 1 1 241 Am 1 1 0.9706 1 1 89.5744 1 1 0.0108 1 1 0.0129 1 1 241 Pu 1 1 102.3696 1 1 35.9113 1 1 2.8506 1 1 2.9619 1 1 239 Pu 1 1 89.4405 1 1 45.7218 1 1 1.9562 1 1 1.7977 1 1 237 Np 1 1 0.5362 1 1 22.5075 1 1 0.0238 1 1 0.0153 1 1 238 U 1 1 0.1171 1 1 3.2031 1 1 0.0366 1 1 0.1195 1 1 235 U 1 1 44.6759 1 1 8.8204 1 1 5.0651 1 1 4.2982 The calculated results were compared with the effective cross sections from [ 43 43 ]. The cross sections from the ENDF reflect the resonance absorption of neutrons around the epithermal energy range. However, for some nuclei such as 238 U, the resonance phenomenon of the reaction cross sections leads to a deviation during the integration process in Eq. (12). However, in [ 43 43 ], the differential cross sections are continuous without showing the resonance phenomenon, which makes the integration results more precise. Because 238 U(n, γ ) plays an important role in the uranium-plutonium cycle, we use the fission-to-capture ratio in [ 43 43 ] to calibrate the capture cross section of 238 U. 3.5 Operational parameters of nuclear power plant 2 The characteristics of the Daya Bay nuclear power plant were adopted [ 64 64 ], and some characteristics are listed in Table 6 Table 6 . It is assumed that the reactor operates at a full power of 330 days per year. The energy production is assumed to be constant. In this study, the initial enrichment is assumed to be 3%, and the fuel cycle performance with 235 U enrichments of above 5% was studied in [ 65 65 ]. The moderator temperature was assumed to be 290°C. The neutron flux was determined using the following equation: 1 1 Parameters 1 1 Designed value 1 1 Actual value 1 1 Maximum continuous electrical power (MW) 1 1 984 1 1 984 1 1 Rated thermal power (MW) 1 1 2905 1 1 2905 1 1 Thermal efficiency (%) 1 1 33.87 1 1 34.1 1 1 Initial enrichment (%) 1 1 1.8/2.4/3.1 1 1 1.8/2.4/3.1 1 1 Initial total inventory of uranium (t) 1 1 72.14 1 1 72.14 1 1 Fuel assembly model 1 1 AFA-2G 1 1 AFA-2G 1 1 Coolant inlet temperature (°C) 1 1 292.4 1 1 293.5 1 1 Coolant outlet temperature (°C) 1 1 327.6 1 1 326.5 <math display="block" id="M33"><mrow><mi>P</mi><mo>=</mo><mstyle displaystyle="true"><msub><mo>∑</mo><mi>f</mi></msub></mstyle><mi>Φ</mi><mi>V</mi><msub><mi>E</mi><mi>f</mi></msub><mn>,</mn></mrow></math> where P is the thermal power of the reactor, ∑ f denotes the macroscopic fission cross section, Φ is the neutron flux, and E f represents the energy released in one fission event of 235 U. In addition, V is the volume of the reactor core. In this study, it is considered that the energy released in one fission event of 235 U is 200 MeV. In general, decreasing the fission cross section leads to an increase in the neutron flux as the fuel consumption increases. 3.6 Long-lived heavy nuclides in PWR 2 In this section, the inventory of long-lived heavy nuclides as a function of fuel consumption is studied. The fuel consumption was calculated using the following equation: <math display="block" id="M34"><mrow><msub><mi>B</mi><mtext>U</mtext></msub><mo>=</mo><mfrac><mrow><mstyle displaystyle="true"><mrow><msubsup><mo>∫</mo><mn>0</mn><mi>T</mi></msubsup></mrow></mstyle><mi>P</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mtext>d</mtext><mi>t</mi></mrow><mrow><msub><mi>M</mi><mtext>U</mtext></msub></mrow></mfrac><mn>,</mn></mrow></math> where B U is the uranium burnup, P represents the thermal power as a function of time, T indicates the total operation time of the reactor. According to [ 68 68 ], a reactor in Daya Bay nuclear plant goes through a shutdown and refueling process every 1 and a half years. Each refueling process lasted for approximately 40 days. In addition, the reactor operates normally at full power on other days. Therefore, for actual fuel utilized for a 1-year period and fuel burned incessantly under full power for approximately 330 days, it is assumed in this study that the burnup in both cases is equivalent. As a result, for a time scale of 1 year, the numerator of Eq.(21) can be expressed as the maximum and constant thermal power multiplied by 330 days. In addition, M U indicates the total mass of uranium in the fuel. Figure 10 Figure 10 illustrates the evolution of inventories of uranium isotopes with burnup. It is noted that for the isotopes of uranium, the inventory of 238 U remains essentially constant. Here, (n, γ ) is the primary consumption path of 238 U owing to the small effective microscopic fission cross sections. With a gradual decrease in the slope, the value of 235 U starts to stabilize at deep burnup after 180 GW ⋅ d/t. Compared to the inventory of 235 U, 234 U and 236 U are insignificant under a low fuel consumption before reaching 50 GW ⋅ d/t. However, the inventory of 236 U exceeds that of 235 U after the burnup at approximately 50 GW ⋅ d/t. The inventory of 234 U is higher than that of 235 U at a deeper burnup. The inventory of 233 U is maintained at an extremely low level after reaching a steady state. Partitioned from the PWR, plutonium can be reutilized in the fuel transition to the thorium fuel cycle in a thermal molten salt reactor [ 66 66 ]. The variation in plutonium isotope inventories with burnup is shown in Fig. 11 Fig. 11 . It is observed that the inventory of 239 Pu is higher than that of several isotopes. The significant generation of 239 Pu at low burnup is the result of a large amount of neutron capture by 238 U and the relatively short half-lives of β -decay of 239 U and 239 Np. In addition, the 240 Pu and 241 Pu levels were maintained at essentially the same level after stabilization. The inventory of 242 Pu increases with the burnup and approaches that of 239 Pu. In Ref. [ 28 28 ], the calculated results in a thermal molten salt reactor using Th-U mixed fuel reflect the same relative inventory of 242 Pu. The main source of 242 Pu is the orbital electron capture of 242 Am. Its relatively short half-life results in a significant inventory of 242 Pu at a deeper burnup. In Fig. 12 Fig. 12 , the inventories of neptunium isotopes are presented. It can be seen that both 237 Np and 238 Np stabilized after burnup at approximately 80 GW ⋅ d/t. The inventories of the americium and curium isotopes are presented in Figs. 13 Figs. 13 and 14 14 . The inventories of the light isotopes are relatively high for both americium and curium isotopes. The heavier isotopes are essentially produced by the neutron capture reaction of lighter isotopes. In addition, some isotopes tend to have the same value at a deeper burnup. The inventory of 243 Am was close to that of 242 Am. The same phenomenon was observed for pairs of ( 243 Cm, 244 Cm) and ( 245 Cm, 246 Cm). This can be explained by comparing the microscopic fission cross sections shown in Table 7 Table 7 . For nuclei such as 244 Cm, 246 Cm, and 243 Am, their fission cross sections are much smaller than those of other isotopes, which leads to less consumption and gradual accumulation. 1 1 Nuclide 1 1 σ f (barns) 1 1 σ c (barns) 1 1 246 Cm 1 1 0.5213 1 1 1.369 1 1 245 Cm 1 1 129.5611 1 1 20.7079 1 1 244 Cm 1 1 0.832 1 1 7.6435 1 1 243 Cm 1 1 62.0526 1 1 11.2636 1 1 243 Am 1 1 0.4443 1 1 27.3167 1 1 242 Am 1 1 200.1696 1 1 21.5212 Figure 15 Figure 15 presents the variation of inventories of uranium and plutonium isotopes with the burnup. It can be seen that the calculated results of the relative inventories and the variation trend in our model are in surprisingly good agreement with the experimental data from Daya Bay. Further investigation is therefore justified, although the calculated absolute inventory of each isotope is higher than the experimental value because the enrichment of uranium used in our simulations was higher than the fuel used in [ 69 69 ], whereas the assumed neutron flux of the reactor is kept at a relatively high level. The total macroscopic fission cross sections and the principal contribution of fission cross sections of fissile nuclides are presented in Fig. 16 Fig. 16 . It was observed that the total fission cross section continued to decrease during the reactor operation. At a lower burnup, it decreases with a slower rate of change, which is mainly caused by a rapid consumption of 235 U, alleviated by the growth of 239 Pu and 241 Pu. With the stabilization of the 239 Pu and 241 Pu inventory, the sustained decrease of 235 U leads to a decrease in the total macroscopic fission cross sections. Typically, PWR nuclear power plants have a discharge burnup of over 45 GW ⋅ d/t. Table 8 Table 8 shows the predictions of the inventory of heavy long-lived nuclides at 50 GW ⋅ d/t and 196 GW ⋅ d/t under full-power operation. 1 1 Nuclide 1 1 Half-life (year) 1 1 Inventory (kg/t) 1 1 1 1 1 1 1 1 50 GW ⋅ d/t 1 1 196 GW ⋅ d/t 1 1 246 Cm 1 1 4723 1 1 6.13×10 -7 1 1 1.92×10 -4 1 1 245 Cm 1 1 8250 1 1 3.85×10 -6 1 1 1.28×10 -4 1 1 244 Cm 1 1 18.11 1 1 1.24×10 -4 1 1 0.0026 1 1 243 Cm 1 1 28.9 1 1 8.73×10 -4 1 1 0.0025 1 1 242 Cm 1 1 0.4459 1 1 0.036 1 1 0.060 1 1 243 Am 1 1 7367 1 1 6.81×10 -5 1 1 2.46×10 -4 1 1 241 Am 1 1 432.6 1 1 0.068 1 1 0.069 1 1 242 Pu 1 1 3.73×10 5 1 1 1.67 1 1 6.67 1 1 241 Pu 1 1 14.33 1 1 2.17 1 1 2.41 1 1 240 Pu 1 1 6561 1 1 2.90 1 1 3.03 1 1 239 Pu 1 1 2.41×10 4 1 1 7.17 1 1 7.28 1 1 237 Np 1 1 2.14×10 6 1 1 0.36 1 1 0.60 1 1 238 U 1 1 4.47×10 9 1 1 978.77 1 1 976.81 1 1 236 U 1 1 2.34×10 7 1 1 4.21 1 1 3.06 1 1 235 U 1 1 7.04×10 8 1 1 2.94 1 1 0.0060 1 1 234 U 1 1 2.46×10 5 1 1 0.001 1 1 0.017 1 1 233 U 1 1 1.59×10 5 1 1 8.78×10 -8 1 1 2.30×10 -7 1 1 231 Pa 1 1 3.27×10 4 1 1 4.10×10 -13 1 1 7.17×10 -15 1 1 232 Th 1 1 1.40×10 10 1 1 2.96×10 -7 1 1 1.14×10 -6 1 1 227 Ac 1 1 21.772 1 1 4.03×10 -13 1 1 7.05×10 -15 1 1 226 Ra 1 1 1600 1 1 6.26×10 -19 1 1 2.75×10 -18 3.7 Fission fragments in a PWR 2 As shown in Table 9 Table 9 , we simulated the nuclide inventory on three decay chains. According to the data provided in [ 70 70 ], the relative inventories of long-lived nuclides are reasonable. Fig.17 Fig.17 presents the evolution of several long-lived nuclides in a PWR under 100% power. As can be observed in Fig.17 Fig.17 , long-lived nuclides increase linearly during the evolution. This accumulation is due to the long half-lives. A few long-lived nuclides can be consumed within a fuel cycle. 1 1 Nuclide 1 1 Half-life(year) 1 1 Inventory(kg/t) GW ⋅ d/t 1 1 90 Br 1 1 6.09 × 10 -8 1 1 1.29 × 10 -9 1 1 90 Kr 1 1 1.02 × 10 -6 1 1 1.94 × 10 -7 1 1 90 Rb 1 1 5.01 × 10 -6 1 1 1.32 × 10 -6 1 1 90 Sr 1 1 28.90 1 1 3.18 × 10 -1 1 1 107 Mo 1 1 1.10 × 10 -7 1 1 2.00 × 10 -9 1 1 107 Tc 1 1 6.72 × 10 -7 1 1 7.47 × 10 -8 1 1 107 Ru 1 1 7.13 × 10 -6 1 1 1.05 × 10 -6 1 1 107 Rh 1 1 4.12 × 10 -5 1 1 6.12 × 10 -6 1 1 107 Pd 1 1 6.50 × 10 6 1 1 1.16 × 10 -1 1 1 135 Sn 1 1 1.68 × 10 -8 1 1 8.88 × 10 -13 1 1 135 Sb 1 1 5.32 × 10 -8 1 1 7.18 × 10 -10 1 1 135 Te 1 1 6.02 × 10 -7 1 1 1.67 × 10 -7 1 1 137 Te 1 1 7.89 × 10 -8 1 1 2.32 × 10 -9 1 1 135 I 1 1 7.50 × 10 -4 1 1 5.89 × 10 -4 1 1 137 I 1 1 7.77 × 10 -7 1 1 3.27 × 10 -7 1 1 135 Xe 1 1 1.04 × 10 -3 1 1 8.60 × 10 -4 1 1 137 Xe 1 1 7.26 × 10 -6 1 1 5.86 × 10 -6 1 1 135 Cs 1 1 2.30 × 10 6 1 1 8.47 × 10 -1 1 1 137 Cs 1 1 30.08 1 1 8.56 × 10 -1 On the one hand, Table 9 Table 9 indicates that the short-lived nuclides are maintained at an extremely low level. For example, the inventory of 135 Sn is approximately 10 -13 kg/t. On the other hand, the long-lived nuclide 135 Cs is greater than the others, and during a 600-day period, increases linearly. For the isobars of 135 Cs, it can be seen that their concentration does not increase significantly because of their short half-life; they increase slowly and start to stabilize at a deeper burnup. Specifically, the inventories of 135 Xe and 135 I are all 10 -4 kg/t at 20 GW ⋅ d/t. 4 Summary 1 As an important aspect of the study on the fission mode and fission fragment inventories in nuclear reactors, fragment yields of fissile nuclides have been investigated for decades. The five Gaussian functions and the most probable charge model were combined to investigate the fission yields in a PWR. The potential energy surface based on the di-nuclear system concept was applied to evaluate the physical properties of the fission process. In addition, the neutron incident energy effects on the fission product distribution are considered. The neutron spectrum in a PWR under stabilization was established using a piecewise function. The inventories of long-lived heavy nuclei and fission fragments in the PWR were predicted by solving a set of differential equations coupled to multiple variables. In this study, mathematical expressions for the neutron energy spectrum in a PWR were established, and effective microscopic cross sections were calculated using data from the ENDF library. The operating parameters of the Daya Bay nuclear power station were used to simulate the inventories of heavy nuclei and fission fragments. For the heavy nuclei, the evolution of uranium, plutonium, neptunium, americium, and curium isotopes were investigated. The inventories of 234 U and 236 U exceed that of 235 U after reaching a discharge burnup. Among the plutonium isotopes, the inventory of 242 Pu increases and gradually reaches the same level as 239 Pu under a deep burnup. Because of the single decay chain for americium and curium isotopes and their large difference in effective fission cross sections, the inventories of the isotopes decrease with an increase in mass number. By contrast, isotope pairs such as 243 Am and 242 Am tend to have similar inventories after stabilization. Upon discharge burnup, except for 238 U, the inventories of 239 Pu, 240 Pu, 241 Pu, 242 Pu, 237 Np, 235 U, and 236 U are predominant in a PWR. For fission fragments, the evolution of several long-lived fission fragments is predicted, and inventories of isobars of 135 Xe were calculated. We also compared the calculated inventories of uranium and plutonium isotopes with experimental data from Daya Bay. Surprisingly, the evolution trends and relative values were in good agreement with the data. 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Current Issues in Nuclear Data Evaluation Methodology: 235U Prompt Fission Neutron Spectra and Multiplicity for Thermal Neutrons All authors contributed to the study conception and design. Material preparation, data collection and analysis were performed by Ze-Xin Fang, Meng Yu and Long Zhu. The first draft of the manuscript was written by Ze-Xin Fang and Meng Yu. All authors commented on previous versions of the manuscript. All authors read and approved the final manuscript. The authors thank Dr. Li-Le Liu and Chao-Sheng Zhou for their useful discussions. 10.1007/s41365-021-00911-0 This work was supported by the National Natural Science Foundation of China (No. 11605296) and the Natural Science Foundation of Guangdong Province, China (No. 2016A030310208) , and Fundamental Research Funds for Central Universities (No. 18lgpy87) . 2021-03-15 2021-05-07 2021-05-13 2021-07-14 2021-07-01 2021-08-05 © China Science Publishing & Media Ltd. (Science Press), Shanghai Institute of Applied Physics, the Chinese Academy of Sciences, ChineseNuclear Society and Springer Nature Singapore Pte Ltd. 2021 This is a post-peer-review, pre-copyedit version of an article published in Nuclear Science and Techniques. The final authenticated version is available online at: http://dx.doi.org/10.1007/s41365-021-00911-0 http://dx.doi.org/10.1007/s41365-021-00911-0 . 2021 Nuclear Science and Techniques 07 32