Independent fission yield data and covariance generation

The independent fission yield (IFY) Y_I(A, Z, M is required in decay heat calculations, which represents the probability of a particular nucleus with mass A, charge Z and isomeric state M being produced directly from one fission after the emission of prompt neutrons and photons but before the emission of delayed neutrons. The most commonly used general-purpose nuclear data libraries: ENDF/B, JENDL and JEFF, provide IFY data with uncertainties as standard deviations. ENDF/B-VIII.0 takes binary fission only, whereas JENDL-5 and JEFF-3.3 both include the light products of ternary fission. To date, no correlation between the fission yield has been reported in these libraries.

The IFY data in these libraries are generally provided by empirical models that are constrained by physical conditions and conservation equations, reflecting the general nature of a fissioning system. These properties should hold in the covariance generation for IFYs, corresponding to the constraint conditions in the GLS updating process. In the present work, the following constraints are employed:

(1) Mass and charge conservation

In a real fission event, the compound nucleus (CN) is generally split into two fission fragments in a process called binary fission. These fragments decay into fission products (FPs) by releasing neutrons and photons. Furthermore, ternary fission may occur with a low probability (~0.2–0.5%), producing extra light-charged particles (LCP), such as protons, triton and α particles [31]. Notably, light-charged particles were also included in the fission products in this study, and the yield data of LCPs and relatively heavy fission products would be updated together within the GLS updating process.

Conservation laws state that the mass and charge numbers of the compound nucleus must be conserved as follows: (1) (2)where A_CN and Z_CN represent the mass and charge of the compound nucleus, respectively. The mass number, charge number, and isomeric state of ith FP are Ai, Zi and Mi respectively. The independent yield is . A, Z and Y_I are the vectors containing the mass number, charge number, and IFY of each FP. The superscript t denotes the transposition of a vector or matrix. is the average number of prompt neutrons. Ai ranges from 66 to 172 for IFYs in ENDF/B-VIII.0, because only binary fission was considered in this library. The range of Ai will be expanded for the yield data in JENDL-5 and JEFF-3.3 to include light-charged particles. Similar phenomena occur in the charge numbers of the fission products Zi.

(2) Normalization of IFYs

The sum of IFYs for FPs except LCPs should be 2 by definition, that is,. (3)The sensitivity vector H₁ is a unit vector for ENDF/B-VIII.0, whereas it has zero elements for LCPs in JENDL-5 and JEFF-3.3.

(3) Normalization of heavier mass yields:

For a set of IFY data, there should be a midpoint mass number A_mid such that the yields of FPs on either side of this midpoint are equal to one when the yield data of LCPs are not counted. As there is a normalization constraint of IFYs to be 2, this condition can be expressed without repetition as (4)where H₂ is a sensitivity vector with unit coefficients for products with mass numbers A greater than A_mid and zero elsewhere. In fact, Eq. (4) is an approximate condition because the mass number is an integer rather than a continuous number. Hence, A_mid is set to .

(4) Charge symmetry

In low-energy neutron-induced fission, FP has a low probability of releasing charged particles during its prompt de-excitation. Therefore, the charge distribution of IFYs should be strictly symmetrical if only binary fission occurs, whereas this symmetry is slightly broken by ternary fission. Following the procedure of Mills [32], the charge symmetry condition was applied to the charge number pairs (Z,Z_CN-Z) with relatively large yields: (5)where H₃ is a sensitive vector defined asfollows: (6)In practical calculations, this constraint is used when the charge yields Y_I(Z) and Y_I(Z_CN-Z) are greater than 1%. .

(5) Chain yield constraints:

The chain yield (ChFY), Y_Ch(A, is defined as the sum of the cumulative yields of the last stable nuclei with the same mass number A and is much more precise than IFY experimental data. A strong negative correlation between different nuclides occurs if ChFY is used as a constraint in the evaluation of IFY.

As noted by England and Rider [33], the chain yields were evaluated after both prompt and delayed neutron emissions. Therefore, the pre-delayed neutron mass yields of IFYs must be corrected to post-delayed neutron chain yields. Based on this definition, ChFY can be calculated as follows: (7)where Y_C denotes cumulative yield. bij is the branching ratio at which the jth nucleus decays to the ith nucleus. di=1 if the ith nucleus is stable, and di=0 otherwise. Formally, Eq. (7) can be written in matrix form Y_Ch=D^tY_I to describe the relationship between the chain yields and independent fission yields. D^t=d^t[E-b]^-1, E denotes the unit matrix. d and b are the vector and matrix filled with elements di and bij respectively.

Currently, there are thousands of radionuclides in the evaluated nuclear data libraries, and their half-lives T_1/2 vary from ~10^-20 to 10²⁰. However, not all radionuclides should be considered in ChFY calculations. The chain yields are close to the mass distribution of the independent fission yields and their differences result from crossing-mass-chain decay processes in which the parent and daughter nuclides are in different mass chains. In the actual measurement of ChFY, the crossing mass chain decay pathway will have almost no contribution to the divergence between Y_Ch(A and Y_I(A if the half-life of the parent nuclide is too long. Hence, choosing a suitable cutoff value for the half-life T_cut is practical. The decay process of a radionuclide was considered only when its half-life was less than T_cut. Otherwise, the radionuclide is regarded as a stable nucleus. Notably, this half-life cutoff is only valid for ChFY calculations, which implies that the influence of the crossing mass-chain decay processes of radionuclides with T_1/2 > T_cut on the chain yields is negligible. These relatively long-lived radionuclides can still influence the time evolution of the fission product composition and the true value of cumulative yields.

Figure 1 shows the calculated ChFYs for the thermal neutron-induced fission of ²³⁵U using the fission yield and decay data in ENDF/B-VIII.0. When T_cut=0, all decay processes were eliminated in ChFY calculations, and the result was equivalent to the mass distribution of the independent fission yields. With increasing T_cut, the calculated Y_Ch(A) departs from Y_I(A gradually and converges at T_cut=1 min. The relative divergence between the converged Y_Ch(A and Y_I(A can reach ~10%, mainly located in the mass regions A=83–97 and 135–140. These differences were primarily attributed to the crossing mass-chain decay processes of radionuclides with half-lives between 1 ms and 1 min. The decay processes of nuclides with T_1/2>1 min hardly changed in the chain yields. Hence, T_cut=1 min is adopted in the following calculations: This cutoff is consistent with the evaluation of England and Rider [33], in which the half-lives of the involved delayed neutron precursors were no longer than 1 min.

Fig. 1Full-screen View

(Color online) (a) Calculated chain yields for thermal neutron-induced fission of ²³⁵U with different half-life cutoff T_cut, using the fission yield and decay data in ENDF/B-VIII.0. The result with T_cut=0 is equivalent to the mass distribution of IFYs Y_I(A. Solid, dashed, dotted, dash-dotted, short dashed and short dotted lines represent the results calculated with T_cut=0, 1 ms, 1 s, 10 s, 1 min and 10 min respectively. (b) The ratio of ChFY calculated with non-zero T_cut to Y_I(A. See text for details

The generalized least squares method [16, 28] was adopted to estimate the covariance matrices of IFYs. For a certain fissioning system, the independent fission yields in the evaluated library are regarded as prior knowledge of the parameters θ_a=Y_I, along with a diagonal prior covariance matrix V_a whose diagonal elements are taken from the corresponding variances. The constraints above can be summarized as the observation equation η~y_a=S^t·θ_a, where η is a vector and S denotes the design matrix, In the GLS approach, the prior parameters and covariance can be updated asfollows: (8)where θ_upd and V_upd are the posterior best estimates and covariance for IFYs respectively. V denotes the covariance matrix of η;

In this study, V is also a diagonal matrix with diagonal elements obtained from the variances of η. The uncertainties of and chain yield were used directly for the mass conservation and chain yield constraints, and fairly small errors were set for the other constraints: 10^-6 for normalization, 0.5% for charge symmetry, in the order of the probability of ternary fission, and 0.01% for the rest. Notably, this study focuses on fission yield covariance generation and its application in decay heat uncertainty analysis, rather than fission yield data evaluation. Hence, and its uncertainty in each library were used in the GLS updating procedure for IFYs in the corresponding library. Similarly, the chain yield data of England and Rider [33] were used as constraints for IFYs in ENDF/B-VIII.0, JENDL-5, whereas those of Nichols et al. [34] were used for IFYs in JEFF-3.3, as indicated in Refs. [9, 17, 35]. Uncertainties in branching ratios were not considered in the chain yield constraints.

Table 1 lists the impact of GLS updating process on each constraint for the thermal neutron-induced fission of ²³⁵U. The residual errors x=η–S^t·θ and their uncertainties are calculated for the observation equations Eqs. (1),(2),(3), and(4), where V_θ denotes the covariance matrix of θ. The most significant effect is the reduction in the uncertainties of the residual errors by several orders of magnitude, particularly for the mass and charge conservation constraints. In most cases, the residual errors also decreased after GLS updating process.

For the charge symmetry and chain yield constraints, the quantity χ² was adopted to quantitatively assess the improvement, which is defined as (9)where N is the length of the vector η. A strong reduction in χ² was observed in most cases, except for the charge symmetry constraint for IFY data in JEFF-3.3. This is because the independent yields in JEFF-3.3 [35] were adjusted to ensure the equality of the complement charge yields with higher precision than in the present work. However, this did not affect the effectiveness of GLS updating procedure. Figure 2 shows the original and updated complementary charge yields as well as Y_I(Z)-Y_I(Z_CN-Z) for ²³⁵U(n_th,f). For the fission yield data in ENDF/B-VIII.0 and JENDL-5, GLS updating procedure slightly adjusts IFYs to reduce the deviations between Y_I(Z) and Y_I(Z_CN-Z) and generates covariance matrix to decrease the uncertainties of Y_I(Z) and Y_I(Z)-Y_I(Z_CN-Z). For JEFF-3.3, there is a noticeable reduction on the uncertainties of Y_I(Z) and Y_I(Z)-Y_I(Z_CN-Z), while Y_I(Z) themselves remain almost unchanged. Similarly, an improvement in agreement with the chain yield data also occurs, along with a reduction in uncertainties, as presented in Fig. 3. Notably, the uncertainties in the calculated chain yields were reduced to no greater than those of the evaluation values in all mass regions after GLS updating process.

Fig. 2Full-screen View

(Color online) (Left panel) Comparison of original and updated charge yields Y_I(Z) for ²³⁵U(n_th,f). (Right panel) Comparison of Y_I(Z)-Y_I(Z_CN-Z) calculated with original and updated yield data for ²³⁵U(n_th,f). The black and red circles represent the results calculated with original and updated IFYs. The results are plotted only when Y_I(Z) and Y_I(Z_CN-Z) are both larger than 1%. See text for details

Fig. 3Full-screen View

(Color online) Impact of GLS updating procedure on chain yield for ²³⁵U(n_th,f) based on the data file from ENDF/B-VIII.0. Evaluation data are taken from England and Rider [33]. (a) Comparison of original and updated chain yields as well as the evaluation values. (b) Ratio of calculated results to evaluation values (C/E). (c) Comparison of relative uncertainties

As shown in Eq. (8), GLS updating procedure not only generates the correlation of fission yields but also adjusts the yield data. A comparison of the original IFYs in ENDF/B-VIII.0 with updated IFYs are presented in Fig. 4 for thermal neutron-induced fission of ²³⁵U. The adjustments are visible only when IFYs have high sensitivity to the constraint system. Strong reductions occur in the variances as they are removed from the diagonal part and reintroduced as correlations between IFYs.

Fig. 4Full-screen View

(Color online) Ratio of updated to original data for (a) IFYs Y_I(A, Z, M and (b) their uncertainties for ²³⁵U(n_th,f) based on the data file from ENDF/B-VIII.0. The values are denoted by circles, squares, rhombuses and triangles when the original IFYs (Y_ori) are larger than 10^-3, in the regions 10^-6–10^-3, 10^-9–10^-6 and smaller than 10^-9 respectively. See text for details

Figure 5 presents the correlation matrices for Y_I(A and Y_I(Z) of the updated IFYs based on the data file from ENDF/B-VIII.0. The correlation matrix is connected to the covariance matrix V by: (10)It can be observed that there are mainly negative correlations between Y_I(A and the neighboring mass numbers, and the correlation becomes insignificant with an increase in the difference in mass. Relatively strong negative correlations were observed in the mass regions A=84–85–86, 87–88, and 136–137, which were all located in the regions where the converged Y_Ch differed from Y_I(A, as shown in Fig. 1(b). It can be inferred that these relatively strong negative correlations resulted from the inclusion of crossing-mass-chain decay processes in the chain yield constraint. In contrast to the mass yields, both positive and negative off-diagonal elements exist in the correlation matrix for Y_I(Z), revealing a more complicated dependence structure for the charge yields.

Fig. 5Full-screen View

(Color online) Correlation matrices for (a) Y_I(A and (b) Y_I(Z) of updated IFYs based on the data file from ENDF/B-VIII.0 for ²³⁵U(n_th,f)

Summation calculation and generalized perturbation theory

The fission pulse decay heat is the heat generated by radioactive decay after an instantaneous burst of fission occurring at time t=0. FPDH can be calculated using the summation method asfollows: (11)where λi, and Ni are the decay constant, average released energy per decay, and number of ith fission products, respectively. The released decay energy consists of contributions from the average energy released as the kinetic energy of light-particle transitions (most frequently as β emissions), electromagnetic radiation and heavy particles . The last term has a negligible contribution; therefore, only light particle and electromagnetic decay heat (H_LP and H_EM) calculations were performed in this study.

The time evolution of N(t) follows the Bateman’s equation [36]: (12)IFYs were used as the initial condition N_i(t=0). This ordinary differential equation is solved numerically using the algorithm proposed by Ladshaw et al. [37].

The uncertainty of FPDH was carried out with generalized perturbation theory [29, 30]. The variance of the decay heat VH can be calculated using the sandwich formula asfollows: (13)where SH is the sensitivity vector of the decay heat with respect to the various nuclear data. The sensitivities of the decay heat to the decay constant and decay energy can be expressed analytically as (14)whereas those of the branching ratio and IFY were calculated numerically.

V_σ is the covariance matrix of nuclear data. The covariance of the fission yield was generated in Sect. 2.1. V_σ of the branching ratio is meaningful for the different decay pathways of a single nuclide because of the normalization condition . For ith nucleus, the covariance matrix for the branching ratios is given by (15)where is the uncertainty of branching ratio bji. The derivation of this expression is given in Appendix A. The uncertainty of the branching ratio was set to zero for radionuclides with only one decay pathway. V_σ of the decay constant and decay energy are diagonal matrices with diagonal elements taken from their variances. If the uncertainties of λ, and were not provided in the evaluated library, they were assumed to be 100%.

FPDH results

The light particles and electromagnetic components of FPDH for thermal neutron-induced fission of ²³⁵U are presented in Figs. 6 and 7 respectively, which were calculated with Set-A and Set-B from ENDF/B-VIII.0, JENDL-5, JEFF-3.3. The experimental data were obtained from the IAEA CoNDERC database [38]. The resulting decay heat curves are in good agreement at cooling times above 1000 s, except that JEFF-3.3 provides a relatively large electromagnetic decay heat during 10⁴–10⁵ s. The discrepancies between the calculated results mainly occur within 1000 s and become significant at cooling times shorter than 1 s. In general, all the calculated results at cooling times below 1000 s are within the error range of the experimental data, except for the underestimation of the electromagnetic decay heat for JEFF-3.3 from 5 to 50 s.

Fig. 6Full-screen View

(Color online) (a) Calculated light particle decay heat for ²³⁵U(n_th,f) with Set-A and Set-B. Experimental data are taken from the CoNDERC database [38]. The solid and dashed lines represent the calculated results with Set-A and Set-B respectively. The calculated results based on ENDF/B-VIII.0, JENDL-5, JEFF-3.3 are denoted by black, red and blue lines respectively. (b) Ratio of calculated results to that obtained with ENDF/B-VIII.0 Set-A

Fig. 7Full-screen View

(Color online) Same as Fig. 6 but for electromagnetic decay heat. Experimental data are taken from the CoNDERC database [38]

For both light particles and electromagnetic decay heats, the updated IFYs lead to an enhancement of up to ~5% at cooling times shorter than 1 s and increase the peaks of the curves by approximately 1–2% for ENDF/B-VIII.0 and JENDL-5, whereas the divergences between the calculated results with Set-A and Set-B are small for JEFF-3.3. These differences reveal that the update on IFYs themselves is moderate for ENDF/B-VIII.0 and JENDL-5 but negligible for JEFF-3.3.

Uncertainties of FPDH

The uncertainties in FPDH were propagated from those of the fission yield, decay energy, decay constant, and branching ratio, as formulated in Sect. 2.2. Computations were carried out with both Set-A and Set-B. To perform an uncertainty analysis with the Set-A data, a diagonal covariance matrix is given for the fission yield data, in which the diagonal elements are taken from their variances.

Figures 8 and 9 show the relative uncertainties of the light particles and electromagnetic decay heat for ²³⁵U(n_th,f), including the contributions of each type of nuclear data. The uncorrelated yield data in Set-A provided a ~4% uncertainty in all cases, dominating the total uncertainty for cooling times longer than 100 s. In contrast, the contribution from the yield data is strongly reduced in Set B, and decreases gradually as time evolves, which is a result of the generated covariance matrix.

Fig. 8Full-screen View

(Color online) Relative uncertainties of light particle decay heat for ²³⁵U(n_th,f). The black lines denote the total uncertainty. The red, blue, green and purple lines represent the contributions from fission yield, decay energy, decay constant and branching ratio respectively. The solid and dashed lines represent the results calculated with Set-A and Set-B

Fig. 9Full-screen View

(Color online) Same as Fig. 8 but for electromagnetic decay heat

The uncertainties propagated from the decay energy, decay constant, and branching ratio are almost unchanged in the Set-A and Set-B results because the decay data have not been adjusted. In general, the decay energy was the primary contributor among the three types of decay data. Notably, the decay energy uncertainty obtained with JENDL-5 is much smaller than those obtained with ENDF/B-VIII.0 and JEFF-3.3 at cooling times shorter than 100 s. This is largely because JENDL-5 provides decay energy uncertainties for more radionuclides than ENDF/B-VIII.0 and JEFF-3.3. These radionuclides are primarily short-lived and release energy during early cooling. Therefore, the assumed 100% uncertainties were adopted less for JENDL-5 decay energy data, resulting in a smaller uncertainty.

With the generated covariance of the yield data, the uncertainties of the light particles and electromagnetic decay heat at cooling time 0.1 s are approximately 10% for ENDF/B-VIII.0 and JEFF-3.3, 5% for JENDL-5. For cooling times longer than 10⁵ s, the uncertainties decreased to approximately 1% for all three libraries. The main component of the uncertainty for the light particle decay heat is the decay energy uncertainty. This situation indicates that the electromagnetic decay heat uncertainty obtained with ENDF/B-VIII.0 and JEFF-3.3, while the uncertainties of yield data and decay energy dominate the total uncertainty of electromagnetic heat at cooling times shorter and longer than 50 s, respectively, for JENDL-5.

Contributions of important fission products and their sensitive coefficients

Because GLS updating procedure generates covariance and adjusts IFYs simultaneously, we are concerned about its influence on the contributions of important fission products to decay heat, as well as their sensitivity coefficients. The analysis was performed at a cooling time of 10 s, which is the position of the curve peak for the light-particle heat, as shown in Fig. 6. Moreover, an underestimation was given by JEFF-3.3 for the electromagnetic heat around this time, as shown in Fig. 7.

A comparison of the 40 dominant contributors is shown in Fig. 10 for the light particle decay heat. These nuclides were sorted according to their contributions obtained using the ENDF/B-VIII.0 Set-A. In all cases, over 80% of the light-particle decay heat was provided by these 40 nuclides at the cooling time of 10 s. In contrast to the reasonable agreement in H_LP, visible divergences occurred among the contributions of important fission products obtained using the different evaluated libraries. In particular, the contributions of ¹⁰²Nb, ^96mY, and ^146mLa calculated using JENDL-5 were less than half of those calculated with ENDF/B-VIII.0 and JEFF-3.3, no matter the set of data used. In addition, the adjustment of IFYs has a weak influence on the contributions of ENDF/B-VIII.0 and JEFF-3.3, whereas the contributions of some specific nuclides are clearly affected by JENDL-5, such as ⁹⁷Y, ¹³⁵Te, ^137,138I, and ⁸⁹Br.

Fig. 10Full-screen View

(Color online) Dominant contributors to light particle decay heat of ²³⁵U(n_th,f) at cooling time 10 s. (a) Contributions of important fission products calculated with different sets of data. (b) Ratio of contribution of important fission product to that obtained with ENDF/B-VIII.0 Set-A. Nuclide is denoted in the form Z-[element name]-A or Z-[element name]-A-M if it is in the ground state or isomeric state(see text for details)

More significant divergences were observed in the electromagnetic decay heat, as shown in Fig. 11. These 40 nuclides were the most significant contributors to the calculations using the ENDF/B-VIII.0 Set-A, and their total contribution accounted for over 75% of the electromagnetic decay heat in all cases. Notably, ⁹⁸Zr does not contribute to electromagnetic heat if the decay data of JENDL-5 or JEFF-3.3 are used, whereas it contributes 0.8% in the calculations with ENDF/B-VIII.0. This is because of ⁹⁸Zr is set to zero in JENDL-5 and JEFF-3.3, whereas it is set to 0.449 MeV in ENDF/B-VIII.0. The contributions calculated with JEFF-3.3 are lower than those obtained with the other two libraries for many nuclides, resulting in an underestimation of the electromagnetic heat around the cooling time 10 s. Comparing the results obtained with Set-A and Set-B, the influence of IFYs adjustment is only considerable for JENDL-5, such as ¹³⁸I, ¹³⁶Te, and ⁸⁹Br.

Fig. 11Full-screen View

(Color online) Same as Fig. 10 but for electromagnetic decay heat

The relative sensitivity coefficient of the decay heat (H) to nuclear data (x) is defined as (16)which is slightly different from SH given in Eq. (14). This is discussed in Sect. 3.2, the decay heat uncertainties propagated from the decay data were almost unchanged in the Set-A and Set-B calculations. Therefore, we focused on the sensitivity coefficient relative to IFYs, that is, sH(Y_I).

The compositions of sH(Y_I) are listed in Tables 2 and 3 for the important fission products presented in Figs. 10 and 11 respectively. The nuclide is omitted if its sH(Y_I) is smaller than 0.01 in all calculations. In general, ¹⁰⁰Zr and ⁹³Rb had the highest relative sensitivity coefficients for light particles and electromagnetic decay heat, respectively, among the three evaluated libraries. The relative changes owing to the yield adjustment were smaller than 20% for sH(Y_I) for the two types of decay heat in most cases, revealing a weak influence. However, dramatic changes were observed in the relative sensitivity coefficients of the electromagnetic decay heat calculated using the JENDL-5. For example, the coefficient of ^96mY is enhanced by approximately five times after GLS updating process, whereas those of ¹⁰²Nb and ¹⁴⁶La are reduced by approximately 95% and 65%, respectively.