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

NUCLEAR ENERGY SCIENCE AND ENGINEERING

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

Ze-Xin Fang，

Meng Yu，

Ying-Ge Huang，

Jin-Bei Chen，

Jun Su，

Long Zhu

Nuclear Science and Techniques

Vol.32, No.7

Article number 72

Published in print 01 Jul 2021

Available online 14 Jul 2021

DOI：10.1007/s41365-021-00911-0

37300

Background:

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:

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:

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 ²³⁵U(n,f) and ²³⁹Pu(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:

The calculated isotopic fission yields were in good agreement with the experimental data. Except for ²³⁸U, the contents of ²³⁹Pu, ²⁴⁰Pu, ²⁴¹Pu, ²⁴²Pu, ²³⁷Np, ²³⁵U, and ²³⁶U 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 ⁹⁰Sr, ¹⁰⁷Pd,¹³⁵Cs, and their isobars ⁹⁰Rb, ¹⁰⁷Rh, and ¹³⁵Xe 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:

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.

RadiotoxicityPWRFive GaussiansLong-lived nuclidesFission fragments yields

1 Introduction

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], 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, 3]. Simulations and calculations of the neutron flux densities have also been extensively studied [4-10].

In recent years, significant progress has been made in the mechanism of nuclear fission [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, 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, 21]. The improved scission-point model describes the charge distributions well [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]. 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] 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]. The semi-empirical method can be useful for determination of the neutron spectrum [26].

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 ²³⁹Pu, ²⁴¹Pu, and ²³³U, can be extracted and reused in fission reactions [27, 28]. However, some fission fragments, such as ⁹⁰Rb, ¹⁰⁷Rh, and ¹³⁵Xe are highly radioactive. The transmutation of minor actinides also contributes to long-term radiotoxicity [29]. The ADS system is flexible for lowering such waste [30], and the storage of spent nuclear fuel should be considered crucial [31]. Therefore, it is necessary to investigate the properties of the fission fragment evolution in a PWR in a theoretical manner.

Fig. 1.

(Color online) Neutron spectrums for thermal, intermediate, and fast reactors.

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

2.1 Multi-Gaussian functions

The Gaussian model was first proposed by Wahl[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]:

y (A) = \sum_{i = 1}^{5} \frac{Y_{i}}{\sqrt{2 π} σ_{i}} \exp {- \frac{{(A - \frac{A_{F}}{2} + Δ_{i} + n t)}^{2}}{2 σ_{i}^{2}}},

(1)

where Yi represents the proportion of each Gaussian component. In addition, σi and Δi are the Gaussian parameters, 2nt 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 $\frac{A_{F}}{2} - n t$ , the relationship between the yields of light fragments and heavy fragments can be written as

y (Z_{L}, A_{L}) = y (Z_{F} - Z_{L}, A_{F} - 2 n t - A_{L}) = y (Z_{H}, A_{H}),

(2)

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 Y₁=Y₅, Y₂=Y₄, Δ₁=-Δ₅, Δ₂=-Δ₄, Δ₃=0, σ₁=σ₅, and σ₂=σ₄. The proportions of these Gaussian components, Yi, are normalized such that $\sum_{i = 1}^{5} Y_{i} = 2$ . All of these Gaussian parameters are considered as functions of the excitation energy E^*.

2.2 Potential energy surface

The parameters Y₂ and Y₃ in Eqs.(1) influence the variation tendency of the peak-to-valley ratio $\frac{y_{peak}}{y_{valley}}$ . 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 shows the potential energy surface of ²³⁵U in the di-nuclear system (DNS) concept, which is defined as follows[34-37]

Fig. 2.

(Color online) Potential energy surface for the reaction ²³⁵U(n,f). The dashed and dotted lines indicate the fragment combinations with the minimum potential energy and the configuration in the symmetry fission, respectively.

\begin{array}{l} U (Z_{i}, N_{i}, R) & = U_{L}^{LD} (Z_{L}, N_{L}) + U_{H}^{LD} (Z_{H}, N_{H}) - U_{CN}^{LD} \\ + δ U_{L}^{shell} (Z_{L}, N_{L}) + δ U_{H}^{shell} (Z_{H}, N_{H}) \\ + V^{C} (Z_{i}, N_{i}, R) + V^{N} (Z_{i}, N_{i}, R), \end{array}

(3)

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 ( $U_{i}^{LD}$ ) and microscopic shell correction ( $δ U_{i}^{shell}$ ) 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].

Relatively low potential energies result in high fission yields of the corresponding fragments[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 $\frac{Y_{2}}{Y_{3}}$ is related to the values of V_mid and V_min, and can be written as

\frac{Y_{2}}{Y_{3}} = f (V_{mid} - V_{min}) .

(4)

2.3 Nuclear charge distribution

To obtain the isotopic fission yield, it is also assumed according to the most probable charge model[41] that the fission yield in the isobaric chains with mass number A follows a Gaussian dispersion:

y (Z, A) = \frac{y_{A}}{\sqrt{2 π} σ_{Z}} \exp (\frac{- {(Z - Z_{p} (A))}^{2}}{2 σ_{Z}^{2}}),

(5)

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], the fission yield of full isotopes for a different fission system is studied. It was found that the $\frac{N}{Z}$ ratio of heavy fragments with the most probable charge is closer to the $\frac{N}{Z}$ value of the compound nucleus. Therefore, we use a linear relation to describe the position of the most probable charge in the isobaric chains:

Z_{p} = k (A - A_{H}) + b .

(6)

Considering the charge conservation and symmetry of the mass distribution, k and b are determined as follows:

k = (Z_{F} - \frac{2 A_{H} Z_{F}}{A_{F}}) / (A_{L} - A_{H}),

(7)

b = \frac{A_{H} Z_{F}}{A_{F}},

(8)

where A_L and A_H represent the mass numbers of light and heavy fragments, respectively. In an isobaric chain,

\sum_{i} y (Z_{i}, A) = y (A),

(9)

where y(A) is the mass distribution and y(Zi, A) is the yield of the fission fragment with proton number Zi in the isobaric chain with mass number A. The determination of parameter σ_Z involves the study of 〈σ_Z〉 and $〈 σ_{Z}^{2} 〉$ 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], for a different fission system, $〈 σ_{Z}^{2} 〉$ 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].

2.4 Prediction of components in spent fuel

The inventory of components in a PWR can be obtained through the following formula[43, 44]:

\begin{array}{l} \frac{d N_{i} (t)}{d t} = \sum_{l} N_{l} (t) \cdot σ_{f, l} \cdot Φ \cdot y_{i, l} + \sum_{k} λ_{k} \cdot K_{p, k} \cdot N_{k} (t) + \\ \sum_{j} σ_{c, j} \cdot Φ \cdot N_{j} (t) - (λ_{i} + σ_{a, i} \cdot Φ) \cdot N_{i} (t), \end{array}

(10)

where Ni 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, λi 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]

σ_{x, r} = \int_{0.001 eV}^{10 MeV} d E_{n} σ_{x, r} (E_{n}) χ (E_{n}) .

(11)

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 (E₀ / E_n), which provides a new integration boundary, which is approximated as 23.03 0.

σ_{x, r} = \int_{0}^{23.03} d u σ_{x, r} (u) [E_{n} χ (E_{n})] (u),

(12)

where E₀ = 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

3.1 Determination of Gaussian parameters

The fitting results of ^233,235,238U(n,f) and ^239,240,241Pu(n,f) are shown with experimental data from [45-55] in Figs. 3 and 4.

Fig. 3.

(Color online) Fitting results for ²³³U(n,f), ²³⁵U(n,f), and ²³⁸U(n,f). The lines represent the curve-fitting results. Black dots with error bars denote the experimental data[52-55].

Fig. 4.

(Color online) Fitting results for ²³⁹Pu(n,f), ²⁴⁰Pu(n,f) and ²⁴¹Pu(n,f). The lines represent the curve-fitting results. Black dots with error bars denote the experimental data[45-51].

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 Y₃ increases exponentially with increasing excitation energy, which results in an increase in the peak-to-valley ratio directly. Under the premise that $\frac{A_{F}}{2} - n t$ is the symmetric axis in the mass distribution, Δ₃ is fixed at 0. Δi and σi are assumed to be constant for a fixed fission system. In addition, 2nt denotes the total number of neutrons emitted, which increases slightly with an increase in the excitation energy.

For uranium, Δi and σi 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:

{\begin{matrix} σ_{i} = p a r 1 (A_{F} - 1) + p a r 2, \\ Δ_{i} = p a r 1 (A_{F} - 1) + p a r 2, \\ n t = (p a r 1 (A_{F} - 1) + p a r 2) E^{*} + p a r 3, \\ Y_{3} = \exp {(p a r 1 (A_{F} - 1) + p a r 2) E^{*} \\ + (p a r 3 (A_{F} - 1) + p a r 4)}, \\ Y_{2} = Y_{3} {(V_{mid} - V_{min}) / (0.055 (A_{F} - 1) - 9.215)}^{5.7}, \\ Y_{1} = 1 - Y_{2} - \frac{Y_{3}}{2}, \end{matrix}

(13)

where par1, par2, par3, and par4 are listed inTable 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 inFig.2. Moreover, E^* is the excitation energy of the fission system.

Values of par1, par2, par3, and par4 for different parameters.

parameters	par1	par2	par3	par4
σ₁	0.421 ± 0.017	-94.818 ± 4.039
σ₂	-0.482 ± 0.041	117.18 ± 9.57
σ₃	0	20
Δ₁	-0.866 ± 0.050	229.12 ± 11.80
Δ₂	-0.895 ± 0.046	228.89 ± 10.73
nt	-0.0033 ± 0.0002	0.837 ± 0.043	0.957 ± 0.004
Y₃(A_F≤236)	-0.00805 ± 0.00001	2.155 ± 0.002	-0.1106 ± 0.0001	20.05 ± 0.03
Y₃(A_F>236)	-0.00250 ± 0.00001	0.851 ± 0.002	-0.0146 ± 0.0003	-2.50 ± 0.05

For the case of plutonium isotopes, because the mass numbers of ²³⁹Pu, ²⁴⁰Pu, and ²⁴¹Pu 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, $\frac{Y_{2}}{Y_{3}}$ can be obtained using the following expression:

\frac{Y_{2}}{Y_{3}} = {(\frac{V_{mid} - V_{min}}{2.9})}^{7.5} .

(14)

The value of Yi depends on the excitation energy. The parameters σi and Δi were chosen to be fixed by changing the excitation energy. The values of the Gaussian parameters for ²³⁹Pu, ²⁴⁰Pu, and ²⁴¹Pu are listed inTable 2.

Fixed Gaussian parameters (σ₄, σ₅, Δ₄, and Δ₅ are assumed to satisfy the conditions σ₄=σ₂, σ₅=σ₁, Δ₁+Δ₅=0, and Δ₂+Δ₄=0)

Nuclide	σ₁	σ₂	σ₃	Δ₁	Δ₂
²³⁹Pu	4	3.5	17.5	23.5	16
²⁴⁰Pu	4	3.5	17.5	23.5	16
²⁴¹Pu	3.5	3.2	17.5	22.9	14.9

3.2 Comparisons with experimental data

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 and 6, the calculated isotope yields from the fitted mass distributions of ²³⁵U(n_th,f) and ²³⁹Pu(n_th,f) were compared with the experimental data[56-59]. Numerical comparisons of the experimental independent and simulated yields for ²³⁵U(n_th,f) and ²³⁹Pu(n_th,f) are shown in Tables 3 and 4. As shown in Tables 3 and 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.

Comparison of experimental independent yield data and simulated yield data for ²³⁵U(n_th,f)

Nuclide	Experimental yield	Simulated yield	Relative difference
⁸³Br	0.00020 ±0.00004	0.00016	-0.210
⁸⁴Br	0.00048 ±0.00007	0.00111	1.313
⁸⁵Br	0.00224 ±0.00009	0.00480	1.143
⁸⁶Br	0.00674 ±0.00024	0.01237	0.835
⁸⁸Br	0.01428 ±0.00026	0.01911	0.338
⁸⁹Br	0.01512 ±0.00052	0.01814	0.200
⁹⁰Br	0.01067 ±0.00044	0.01029	-0.036
⁹¹Br	0.00287 ±0.00021	0.00359	0.251
⁸⁵Kr	0.00039 ±0.00007	0.00010	-0.744
⁸⁶Kr	0.00113 ±0.00009	0.00087	-0.228
⁸⁷Kr	0.00468 ±0.00024	0.00456	-0.026
⁸⁸Kr	0.01769 ±0.00054	0.01465	-0.172
⁸⁹Kr	0.03489 ±0.00042	0.02814	-0.193
⁹⁰Kr	0.04782 ±0.00070	0.03320	-0.306
⁹¹Kr	0.03384 ±0.00107	0.02427	-0.283
⁹²Kr	0.01644 ±0.00077	0.01082	-0.342
⁹³Kr	0.00402 ±0.00022	0.00305	-0.241
⁹⁴Kr	0.00092 ±0.00019	0.00054	-0.414
⁸⁷Rb	0.00041 ±0.00012	0.00004	-0.903
⁸⁸Rb	0.00069 ±0.00014	0.00043	-0.371
⁸⁹Rb	0.00252 ±0.00034	0.00282	0.119
⁹⁰Rb	0.00860 ±0.00065	0.01127	0.310
⁹¹Rb	0.02347 ±0.00092	0.02789	0.188
⁹²Rb	0.03354 ±0.00045	0.04208	0.255
⁹³Rb	0.03167 ±0.00079	0.04021	0.270
⁹⁴Rb	0.01557 ±0.00051	0.02402	0.543
⁹⁵Rb	0.00636 ±0.00024	0.00893	0.404
⁹⁶Rb	0.00115 ±0.00020	0.00211	0.835
⁹⁷Rb	0.00037 ±0.00018	0.00031	-0.170
⁹⁰Sr	0.00081 ±0.00024	0.00014	-0.827
⁹¹Sr	0.00229 ±0.00027	0.00118	-0.485
⁹²Sr	0.00959 ±0.00067	0.00600	-0.374
⁹³Sr	0.02571 ±0.00084	0.01942	-0.245
⁹⁴Sr	0.04506 ±0.00052	0.03928	-0.128
⁹⁵Sr	0.04756 ±0.00056	0.04940	0.039
⁹⁶Sr	0.03701 ±0.00075	0.03958	0.069
⁹⁷Sr	0.01794 ±0.00074	0.01949	0.086
⁹⁸Sr	0.00818 ±0.00046	0.00593	-0.275
⁹⁹Sr	0.00155 ±0.00022	0.00112	-0.277
¹³⁸Xe	0.0473 ±0.0032	0.04865	0.029
¹³⁹Xe	0.048 ±0.003	0.04381	-0.085
¹⁴⁰Xe	0.0355 ±0.0013	0.02477	-0.302
¹⁴¹Xe	0.01360 ±0.00060	0.00868	-0.362
¹⁴²Xe	0.00447 ±0.00020	0.00194	-0.566
¹³⁷Cs	0.00061 ±0.00006	0.00130	1.145
¹³⁸Cs	0.00532 ±0.00021	0.00627	0.179
¹³⁹Cs	0.01430 ±0.00055	0.01912	0.337
¹⁴⁰Cs	0.0284 ±0.0012	0.03661	0.289
¹⁴¹Cs	0.031 ±0.001	0.04345	0.397
¹⁴²Cs	0.02684 ±0.00072	0.03280	0.222
¹⁴³Cs	0.01355 ±0.00037	0.01521	0.123
¹⁴⁴Cs	0.00428 ±0.00010	0.00434	0.014
¹⁴⁵Cs	0.00085 ±0.00003	0.00077	-0.101
¹⁴⁶Cs	0.000076 ±0.000006	0.00009	0.070

Comparison of experimental independent yield data and simulated yield data for ²³⁹Pu(n_th,f)

Nuclide	Experimental yield	Simulated yield	Relative difference
⁸⁹Rb	0.00271 ±0.00088	0.00249	-0.081
⁹¹Rb	0.0136 ±0.0021	0.01834	0.349
⁹²Rb	0.0164 ±0.0028	0.02279	0.390
⁹³Rb	0.0195 ±0.0030	0.01707	-0.125
⁹⁴Rb	0.0097 ±0.0020	0.00759	-0.218
⁹⁵Rb	0.0075 ±0.0018	0.00209	-0.721
⁹⁷Rb	0.00057 ±0.00089	0.000038	-0.934
⁹³Sr	0.02310 ±0.00025	0.02076	-0.101
⁹⁴Sr	0.03168 ±0.00034	0.0318	-0.004
⁹⁵Sr	0.02820 ±0.00031	0.03024	0.072
⁹⁶Sr	0.01927 ±0.00021	0.01771	-0.081
⁹⁸Sr	0.00331 ±0.00013	0.00152	-0.541
⁹⁹Sr	0.000400 ±0.000004	0.00022	-0.445
¹⁰¹Tc	0.00014 ±0.00009	0.000003	-0.980
¹⁰³Tc	0.00241 ±0.00097	0.00043	-0.821
¹⁰⁴Tc	0.00536 ±0.00083	0.00247	-0.539
¹⁰⁵Tc	0.01086 ±0.00251	0.00818	-0.247
¹⁰⁴Ru	0.00004 ±0.00001	0.000006	-0.852
¹⁰⁵Ru	0.00008 ±0.00001	0.00007	-0.154
¹⁰⁶Ru	0.00007 ±0.00003	0.00046	-0.363
¹⁰⁷Ru	0.00253 ±0.00028	0.00179	-0.292
¹⁰⁸Ru	0.00619 ±0.00025	0.00409	-0.339
¹⁰⁹Ru	0.00674 ±0.00046	0.00549	-0.185
¹⁴²La	0.00406 ±0.00006	0.00447	0.101
¹⁴³La	0.01045 ±0.00016	0.01253	0.199
¹⁴⁴La	0.01228 ±0.00024	0.02129	0.734
¹⁴⁵La	0.01796 ±0.00027	0.02148	0.196
¹⁴⁶La	0.00742 ±0.00010	0.01315	0.772
¹⁴⁷La	0.00655 ±0.00011	0.00472	-0.279
¹⁴⁸La	0.00191 ±0.00003	0.00101	-0.471
¹⁴⁵Ce	0.00430 ±0.00008	0.00437	0.016
¹⁴⁶Ce	0.00793 ±0.00019	0.00922	0.163
¹⁴⁷Ce	0.01619 ±0.00031	0.01141	-0.295
¹⁴⁸Ce	0.00920 ±0.00022	0.00838	-0.089

Fig. 5.

(Color online) Normalized isotope yields of fission fragments in ²³⁵U(n_th,f) are compared with the experimental data. The hollow points are the calculated results. The red dashed lines are the guidelines. The black solid points represent the experimental data. Data for light fragments were taken from [59]. Data for fragments with Z = 54 and Z = 55 were obtained from [58] and [57].

Fig. 6.

Normalized isotope yields of fission fragments in ²³⁹Pu(n_th,f) are compared with the experimental data. The hollow points are the calculated results. The red dashed lines are the guidelines. The black solid points represent the experimental data. Experimental data were obtained from [56].

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 ²³⁵U(n_th,f) and ²³⁹Pu(n_th,f), as shown inFig. 7). The experimental data were obtained from [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.

Fig. 7.

(Color online) Normalized charge distribution of the fission fragments in ²³⁵U(n_th,f) and ²³⁹Pu(n_th,f) are compared with the experimental data. Red dots are the calculated results, and red lines are the guidelines. The black solid points are the experimental data taken from [56]

3.3 Neutron spectrum

Based on the Daya Bay nuclear power plant, a neutron spectrum was established by modeling an actual reactor core using MCNP4C[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 10⁵ eV to 10 MeV. Most of these neutrons are emitted before and after fission. The energy of epithermal neutrons is between 0.1 and 10⁵ 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

ϕ = E_{n} \times χ (E_{n}),

(15)

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]. We found 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:

χ (E_{n}) = α_{M}^{2} \exp (- E_{n} α_{M}) E_{n},

(16)

where α_M is related to the temperature of the moderating medium [25]:

α_{M} = \frac{1}{T_{m}} .

(17)

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].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.

Fig. 8.

(Color online) Comparison of the thermal neutron spectrum in the case of a water moderated reactor at 291.15 and 371.15 K. Hollow points are the experimental data [67]. Lines represent the theoretical results calculated in this study.

Within the epithermal neutron energy range, the probability density of the neutron energy distribution follows the $\frac{1}{E_{n}}$ law. To make the transition smoother between the three different regions and more suitable to the results in [60], the probability density in this region is assumed as

χ (E_{n}) = C \frac{1}{{(E_{n})}^{0.9}} + \frac{0.02}{{(E_{n})}^{2.1}} + \frac{7}{10^{7} - E_{n}},

(18)

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 ²³⁵U. The experimental data and curve fitting results of the ²³⁵U fission spectrum are presented in [61, 62]. In [62], the prompt neutron spectrum from thermal neutron-induced fission in ²³⁵U using the recoil proton method was recently measured. In [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 ²³⁵U is considered as a linear combination of the two formulas and can be written as

\begin{array}{l} χ (E_{n}) = C_{1} (ω \sqrt{E_{n}} e^{- E_{n} / E_{M}} \\ + (1 - ω) e^{- E_{n} / a} \sinh \sqrt{b E_{n}}), \end{array}

(19)

where ω denotes the weight, and E_M, a, and b are constants. These were all determined in [63] using the least-squares method. In addition, C₁ 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. InFig. 9, the normalized neutron spectrum in the PWR was compared with the calculated results obtained in [60]. In our mathematical description of the neutron spectrum, the resonance in the epithermal energy range was not considered.

Fig. 9.

The neutron spectrum established in this study is compared with the results simulated by MCNP4C in [60]. The solid line represents the neutron spectrum in this study. Dots refer to the calculations[60].

3.4 Effective cross section in PWR

In 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.

Effective fission (σ_f) and neutron capture (σ_c) cross sections for several heavy nuclides

Nuclide	σ_f(barns)	σ_c(barns)	σ_f/σ_c	σ_f/σ_c in Ref.[43]
²⁴²Cm	1.2005	3.0541	0.3931	0.2653
²⁴¹Am	0.9706	89.5744	0.0108	0.0129
²⁴¹Pu	102.3696	35.9113	2.8506	2.9619
²³⁹Pu	89.4405	45.7218	1.9562	1.7977
²³⁷Np	0.5362	22.5075	0.0238	0.0153
²³⁸U	0.1171	3.2031	0.0366	0.1195
²³⁵U	44.6759	8.8204	5.0651	4.2982

The calculated results were compared with the effective cross sections from [43]. The cross sections from the ENDF reflect the resonance absorption of neutrons around the epithermal energy range. However, for some nuclei such as ²³⁸U, the resonance phenomenon of the reaction cross sections leads to a deviation during the integration process in Eq. (12). However, in [43], the differential cross sections are continuous without showing the resonance phenomenon, which makes the integration results more precise. Because ²³⁸U(n, γ) plays an important role in the uranium-plutonium cycle, we use the fission-to-capture ratio in [43] to calibrate the capture cross section of ²³⁸U.

3.5 Operational parameters of nuclear power plant

The characteristics of the Daya Bay nuclear power plant were adopted [64], and some characteristics are listed inTable 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 ²³⁵U enrichments of above 5% was studied in [65]. The moderator temperature was assumed to be 290°C. The neutron flux was determined using the following equation:

Operating parameters of Daya Bay nuclear power plant

Parameters	Designed value	Actual value
Maximum continuous electrical power (MW)	984	984
Rated thermal power (MW)	2905	2905
Thermal efficiency (%)	33.87	34.1
Initial enrichment (%)	1.8/2.4/3.1	1.8/2.4/3.1
Initial total inventory of uranium (t)	72.14	72.14
Fuel assembly model	AFA-2G	AFA-2G
Coolant inlet temperature (°C)	292.4	293.5
Coolant outlet temperature (°C)	327.6	326.5

P = \sum_{f} Φ V E_{f},

(20)

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 ²³⁵U. 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 ²³⁵U 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

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:

B_{U} = \frac{\int_{0}^{T} P (t) d t}{M_{U}},

(21)

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], 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 illustrates the evolution of inventories of uranium isotopes with burnup. It is noted that for the isotopes of uranium, the inventory of ²³⁸U remains essentially constant. Here, (n, γ) is the primary consumption path of ²³⁸U owing to the small effective microscopic fission cross sections. With a gradual decrease in the slope, the value of ²³⁵U starts to stabilize at deep burnup after 180 GW⋅d/t. Compared to the inventory of ²³⁵U, ²³⁴U and ²³⁶U are insignificant under a low fuel consumption before reaching 50 GW⋅d/t. However, the inventory of ²³⁶U exceeds that of ²³⁵U after the burnup at approximately 50 GW⋅d/t. The inventory of ²³⁴U is higher than that of ²³⁵U at a deeper burnup. The inventory of ²³³U is maintained at an extremely low level after reaching a steady state.

Fig. 10.

(Color online) Calculated inventories of uranium isotopes in PWR as a function of burnup.

Partitioned from the PWR, plutonium can be reutilized in the fuel transition to the thorium fuel cycle in a thermal molten salt reactor [66]. The variation in plutonium isotope inventories with burnup is shown inFig. 11. It is observed that the inventory of ²³⁹Pu is higher than that of several isotopes. The significant generation of ²³⁹Pu at low burnup is the result of a large amount of neutron capture by ²³⁸U and the relatively short half-lives of β-decay of ²³⁹U and ²³⁹Np. In addition, the ²⁴⁰Pu and ²⁴¹Pu levels were maintained at essentially the same level after stabilization. The inventory of ²⁴²Pu increases with the burnup and approaches that of ²³⁹Pu. In Ref. [28], the calculated results in a thermal molten salt reactor using Th-U mixed fuel reflect the same relative inventory of ²⁴²Pu. The main source of ²⁴²Pu is the orbital electron capture of ²⁴²Am. Its relatively short half-life results in a significant inventory of ²⁴²Pu at a deeper burnup.

Fig. 11.

(Color online) Calculated inventories of plutonium isotopes in a PWR as a function of burnup.

In Fig. 12, the inventories of neptunium isotopes are presented. It can be seen that both ²³⁷Np and ²³⁸Np stabilized after burnup at approximately 80 GW⋅d/t. The inventories of the americium and curium isotopes are presented in Figs. 13 and 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 ²⁴³Am was close to that of ²⁴²Am. The same phenomenon was observed for pairs of (²⁴³Cm, ²⁴⁴Cm) and (²⁴⁵Cm, ²⁴⁶Cm). This can be explained by comparing the microscopic fission cross sections shown in Table 7. For nuclei such as ²⁴⁴Cm, ²⁴⁶Cm, and ²⁴³Am, their fission cross sections are much smaller than those of other isotopes, which leads to less consumption and gradual accumulation.

Effective fission (σ_f) and neutron capture (σ_c) cross sections for Cm and Am isotopes.

Nuclide	σ_f(barns)	σ_c(barns)
²⁴⁶Cm	0.5213	1.369
²⁴⁵Cm	129.5611	20.7079
²⁴⁴Cm	0.832	7.6435
²⁴³Cm	62.0526	11.2636
²⁴³Am	0.4443	27.3167
²⁴²Am	200.1696	21.5212

Fig. 12.

(Color online) Calculated inventories of neptunium isotopes in a PWR as a function of burnup.

Fig. 13.

(Color online) Calculated inventories of americium isotopes in a PWR as a function of burnup.

Fig. 14.

(Color online) Calculated inventories of curium isotopes in a PWR as a function of burnup.

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], whereas the assumed neutron flux of the reactor is kept at a relatively high level.

Fig. 15.

(Color online) Comparison of heavy nuclide inventories between (A) experimental data [69] and (B) calculated results in a PWR.

The total macroscopic fission cross sections and the principal contribution of fission cross sections of fissile nuclides are presented inFig. 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 ²³⁵U, alleviated by the growth of ²³⁹Pu and ²⁴¹Pu. With the stabilization of the ²³⁹Pu and ²⁴¹Pu inventory, the sustained decrease of ²³⁵U leads to a decrease in the total macroscopic fission cross sections.

Fig. 16.

Calculated total macroscopic fission cross sections and the principle components of fission cross sections.

Typically, PWR nuclear power plants have a discharge burnup of over 45 GW⋅d/t. 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.

Prediction of heavy long-lived nuclides in a PWR

Nuclide	Half-life (year)	Inventory (kg/t)
		50 GW⋅d/t	196 GW⋅d/t
²⁴⁶Cm	4723	6.13×10^-7	1.92×10^-4
²⁴⁵Cm	8250	3.85×10^-6	1.28×10^-4
²⁴⁴Cm	18.11	1.24×10^-4	0.0026
²⁴³Cm	28.9	8.73×10^-4	0.0025
²⁴²Cm	0.4459	0.036	0.060
²⁴³Am	7367	6.81×10^-5	2.46×10^-4
²⁴¹Am	432.6	0.068	0.069
²⁴²Pu	3.73×10⁵	1.67	6.67
²⁴¹Pu	14.33	2.17	2.41
²⁴⁰Pu	6561	2.90	3.03
²³⁹Pu	2.41×10⁴	7.17	7.28
²³⁷Np	2.14×10⁶	0.36	0.60
²³⁸U	4.47×10⁹	978.77	976.81
²³⁶U	2.34×10⁷	4.21	3.06
²³⁵U	7.04×10⁸	2.94	0.0060
²³⁴U	2.46×10⁵	0.001	0.017
²³³U	1.59×10⁵	8.78×10^-8	2.30×10^-7
²³¹Pa	3.27×10⁴	4.10×10^-13	7.17×10^-15
²³²Th	1.40×10¹⁰	2.96×10^-7	1.14×10^-6
²²⁷Ac	21.772	4.03×10^-13	7.05×10^-15
²²⁶Ra	1600	6.26×10^-19	2.75×10^-18

3.7 Fission fragments in a PWR

As shown in Table 9, we simulated the nuclide inventory on three decay chains. According to the data provided in [70], the relative inventories of long-lived nuclides are reasonable.Fig.17 presents the evolution of several long-lived nuclides in a PWR under 100% power. As can be observed inFig.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.

Prediction of fission products in a PWR

Nuclide	Half-life(year)	Inventory(kg/t) GW⋅d/t
⁹⁰Br	6.09 × 10^-8	1.29 × 10^-9
⁹⁰Kr	1.02 × 10^-6	1.94 × 10^-7
⁹⁰Rb	5.01 × 10^-6	1.32 × 10^-6
⁹⁰Sr	28.90	3.18 × 10^-1
¹⁰⁷Mo	1.10 × 10^-7	2.00 × 10^-9
¹⁰⁷Tc	6.72 × 10^-7	7.47 × 10^-8
¹⁰⁷Ru	7.13 × 10^-6	1.05 × 10^-6
¹⁰⁷Rh	4.12 × 10^-5	6.12 × 10^-6
¹⁰⁷Pd	6.50 × 10⁶	1.16 × 10^-1
¹³⁵Sn	1.68 × 10^-8	8.88 × 10^-13
¹³⁵Sb	5.32 × 10^-8	7.18 × 10^-10
¹³⁵Te	6.02 × 10^-7	1.67 × 10^-7
¹³⁷Te	7.89 × 10^-8	2.32 × 10^-9
¹³⁵I	7.50 × 10^-4	5.89 × 10^-4
¹³⁷I	7.77 × 10^-7	3.27 × 10^-7
¹³⁵Xe	1.04 × 10^-3	8.60 × 10^-4
¹³⁷Xe	7.26 × 10^-6	5.86 × 10^-6
¹³⁵Cs	2.30 × 10⁶	8.47 × 10^-1
¹³⁷Cs	30.08	8.56 × 10^-1

Fig. 17.

(Color online) Evolution of long-lived nuclides during a fuel cycle

On the one hand, Table 9 indicates that the short-lived nuclides are maintained at an extremely low level. For example, the inventory of ¹³⁵Sn is approximately 10^-13 kg/t. On the other hand, the long-lived nuclide ¹³⁵Cs is greater than the others, and during a 600-day period, increases linearly. For the isobars of ¹³⁵Cs, 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 ¹³⁵Xe and ¹³⁵I are all 10^-4 kg/t at 20 GW⋅d/t.

4 Summary

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 ²³⁴U and ²³⁶U exceed that of ²³⁵U after reaching a discharge burnup. Among the plutonium isotopes, the inventory of ²⁴²Pu increases and gradually reaches the same level as ²³⁹Pu 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 ²⁴³Am and ²⁴²Am tend to have similar inventories after stabilization. Upon discharge burnup, except for ²³⁸U, the inventories of ²³⁹Pu, ²⁴⁰Pu, ²⁴¹Pu, ²⁴²Pu, ²³⁷Np, ²³⁵U, and ²³⁶U are predominant in a PWR. For fission fragments, the evolution of several long-lived fission fragments is predicted, and inventories of isobars of ¹³⁵Xe 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. In the future, our group will optimize the calculation of the reaction cross sections and consider more conversions between nuclides, allowing the method to be improved and better predict the inventories of long-lived isotopes.

References

[1]

C.H. Zeng,

235U Fission product radiological calculations in nuclear fuel reactors

. China Journal of Nuclear Science and Engineering 12, 81-86 (1992).(in Chinese)