Decay Branch Ratio Sampling Method with Dirichlet Distribution

Abstract

The decay branch ratio is evaluated nuclear data related to the decay heat calculation in reactor safety analysis. Decay branch ratio data are inherently subjected to the “sum-to-one” constraint, making it difficult to generate perturbed samples while preserving their suggested statistics in a library of evaluated nuclear data. Therefore, a stochastic-sampling-based uncertainty analysis method is hindered in quantifying the uncertainty contribution of the decay branch ratio to the decay heat calculation. In the present work, two alternative sampling methods are introduced, based on Dirichlet and generalized Dirichlet distribution, to tackle the decay branch ratio sampling issue. The performance of the introduced methods is justified by three-branch decay data retrieved from ENDF/B-VIII.0. The results show that the introduced sampling methods are capable of generating branch ratio samples and preserving their suggested statistics in an evaluated nuclear data library while satisfying their inherent “sum-to-one” constraint. These decay-branch-ratio sampling methods are expected to be alternative procedures in conducting stochastic-sampling-based uncertainty analyses of the decay branch ratio in reactor simulations.

1. Introduction

Currently, the best-estimate-plus-uncertainty (BEPU) method is widely used to determine the design and safety margins of reactor systems [1]. The most characteristic aspect of the BEPU method is that it explicitly addresses various existent uncertainty contributors to the simulation results of reactors. These uncertainty contributors commonly find themselves in three categories: namely, mathematical model simplification, numerical solver approximations, and uncertain model input parameters [2]. Among these factors, uncertain model input parameters have gradually became a major uncertainty contributor, given the advances in high-fidelity reactor simulation methods and the improvements in massively parallel computers [3,4]. Uncertainty contributions from uncertain parameters are commonly quantified by a stochastic-sampling-based method [5,6,7] or a sensitivity-based deterministic method [8]. Due to the versatile and non-intrusive nature of the stochastic-sampling-based method, it has been widely adopted in conducting parameter uncertainty analyses of reactor simulations in which non-linearity or multiple responses are existent.

The decay heat uncertainty analysis is one of active research domains in reactor simulation and safety analysis, e.g., in a high-temperature gas-cooled reactor (HTGR) [9]. Recently, many advances have been made in the high-fidelity and efficient calculation of burnup [10] as well as decay heat [11]. The uncertainties of these calculations still need to be properly quantified. In decay heat uncertainty analysis, propagating evaluated nuclear data uncertainty to the calculated decay heat plays a central role in the research activity. Among all the considered nuclear data appearing in the calculation of decay heat, for example, the nuclear cross-section, decay constant, and radioactive decay energy [11], the decay branch ratio is featured for its inherent “sum-to-one” constraint. This constraint is not only bound the values of decay branch ratio in a $\left(0,1\right)$ interval but also restrains its variation inside a simplex whose number of nodes is identical to the number of decay branches. As the decay branch ratio’s mean values and standard deviations are provided in the currently released evaluated nuclear data library, the stochastic generation of branch ratio samples, which could preserve their evaluated statistics and respect their inherent constraint, is still an open issue. This issue hinders the quantification of the uncertainty contributions from the evaluated decay branch ratio in the practice of decay heat uncertainty analysis.

In the previous research, this issue was commonly tackled by adopting the deterministic uncertainty quantification method. In this method, a covariance matrix for the decay branch ratio is first estimated from its “sum-to-one” condition. By assuming the existence of linear dependence between the decay branch ratio and decay heat, the uncertainty contribution from the branch ratio is therefore calculated, with respect to each branch ratio, as the quadric form of the estimated covariance matrix under decay heat sensitivities [11,12,13]. Apart from this, some researchers attempted to use the above estimated branch ratio covariance matrix to stochastically sample branch ratios and quantify its uncertainty contribution via the stochastic sampling method [14]. Although this has been implemented, the quantified uncertainty is still restricted by the approximations made in the estimation of the decay branch ratio covariance.

The purpose of this work is to provide an alternative sampling method for stochastically generating branch ratio samples. Compared with the previously mentioned methods, the introduced method needs no decay branch ratio covariance prepared before-hand. By assigning a Dirichlet distribution to the decay branch ratios, their “sum-to-one” constraint could therefore be retained. Dirichlet distribution is a multivariate generalization of Beta distribution, and it has a wide range of applications [15] in describing the stochastic distribution of fractional random variables subjected to a “sum-to-one” constraint. Readers can find a detailed review regarding the history of Dirichlet distribution in [16]. The potential applications of the present work provide an alternative approach to conducting an uncertainty analysis of a decay branch ratio using a stochastic-sampling-based method in the burnup [17,18,19] and decay heat calculation [20,21] of reactors.

The structure of this work is organized as follows: Section 2 provides a general insight into decay branch ratio sampling process using Dirichlet destitution. Two sampling methods, namely, the DIR sampling method and the gDIR sampling method, are introduced in Section 3. These two methods are justified using data from several decay branch ratios, taken from ENDF/B-VIII.0. The results and discussions are provided in Section 4. Finally, remarks and conclusions are presented in Section 5.

2. Decay Branch Ratio Sampling Process

A radioactive isotope has $K$ decay branches, with the decay branch ratios being denoted as the random vector $B_{1:K}={\left[B_1,B_2,\dots ,B_K\right]}^{\mathrm{T}}$. Here, “T” represents the transpose of the vector. For any $i=1,\dots ,K$, each branch ratio $B_i$ takes its value, $b_i,$ between zero and unity, with the mean and standard deviation represented by ${\mu }_i$ and ${\sigma }_i$, respectively. Furthermore, all branch ratios in $B_{1:K}$ are subject to “sum-to-one” constraint, as formulated in Equation (1). In a stochastic sampling uncertainty analysis, a properly developed decay-branch-ratio sampling procedure requires the assignment of a probability density function (PDF) to $B_{1:K}$, which should not only preserve all its suggested statistics (e.g., mean and standard deviation) but also respect to the “sum-to-one” constraint. Dirichlet distribution is one of those probability distributions which satisfies these requirements.

$${\displaystyle \sum }_{i=1}^K{B}_i=1.0$$

(1)

Dirichlet distribution (DIR) and generalized Dirichlet distribution (gDIR) are multivariate, continuous distributions with all random variates taking values in the $\left(0,1\right)$ interval and subject to the “sum-to-one” constraint [22]. The general insight behind DIR and gDIR is related to the random partition of intervals, which is schematically illustrated in Figure 1. Each uncertain decay branch ratio is iteratively taken as the partitioned residual interval, $r,$ with a randomly selected partition, $u$. Three key ingredients are included in this process:

• The residual interval, $r_i,$ is defined as the remaining part of $\left(0,1\right)$ that is taken by the preceding specified branch ratio summation, which is $r_i=1-{\displaystyle \sum }_{m=1}^{i-1}{b}_m$;
• The partition, $U_i,$ is a random variate following a Beta distribution, with its value, $u_i,$ randomly selected in the interval between zero and unity;
• All the partition values, $U_i,$ are mutually independent.

Figure 1. Schematic illustration of branch ratio random partition (branch ratio value is taken as partitioned interval, $r$, with the partition being randomly selected between zero and unity).

Figure 1. Schematic illustration of branch ratio random partition (branch ratio value is taken as partitioned interval, $r$, with the partition being randomly selected between zero and unity).

With the above definition, the uncertain branch ratio, $B_i,$ is therefore randomly determined as $b_i=u_i\times r_i$. It should be noted that, because of the imposed “sum-to-one” constraint, the last branch ratio is uniquely determined as the residual of all the preceding specified branch ratio values. Therefore, there are actually $K-1$ random branch ratios that are iteratively selected from this process.

3. Dirichlet Distribution Based Decay Branch Ratio Sampling

3.1. Development of DIR Sampling Method

Consider decay branch ratios $B_{1:K}$, which follow $K-1$ variates’ Dirichlet distribution, as in Equation (2), denoted as $\mathrm{Dir}\left({\alpha }_{1:K-1};{\alpha }_K\right)$. Here, $\Gamma \left(·\right)$ is the Gamma function and the distributional parameters ${\alpha }_{1:K}=\left\{{\alpha }_k>0:k=1,\dots ,K\right\}$ are all positive. The last decay branch ratio is taken as $B_K=1-{\displaystyle \sum }_{k=1}^{K-1}{B}_k$ due to the “sum-to-one” condition.

$$f_{B_{1:K-1}}\left(b_{1:K-1}\right)=\frac{\Gamma \left({{\displaystyle \sum }}_{s=1}^K{\alpha }_s\right)}{{{\displaystyle \prod }}_{s=1}^K\Gamma \left({\alpha }_s\right)}{\displaystyle \prod }_{s=1}^{K-1}{b}_s^{{\alpha }_s-1}{\left(1-b_1-\dots -b_{K-1}\right)}^{{\alpha }_K-1}$$

(2)

The goal here is to design sampling procedures from which the sampled decay branch ratios have estimated mean values and standard deviations identical to the statistics suggested in the evaluated nuclear data library. One practical way to achieve this goal is to iteratively sample from the conditional factorization of $\mathrm{Dir}\left({\alpha }_{1:K-1};{\alpha }_K\right)$, shown in Equation (3). If the conditional mean value and standard deviation with respect to each conditional PDF, $f_{B_i|B_{1:i-1}}\left(b_i\right),$ are identical to the suggested statistics of the branch ratio, $B_i$, for any $i=1,\dots ,K-1$, the above-mentioned condition could thus be satisfied.

$$f_{B_{1:K-1}}\left(b_{1:K-1}\right)=f_{B_1}\left(b_1\right)·f_{B_2|B_1}\left(b_2\right)·f_{B_3|B_{1:2}}\left(b_3\right)·\dots ·f_{B_{K-1}|B_{1:K-2}}\left(b_{K-1}\right)$$

(3)

Here, $f_{B_i|B_{1:i-1}}\left(b_i\right)$ is the conditional PDF for the decay branch ratio, $B_i,$ given the already known values of the preceding $i-1$ branch ratios, $B_{1:i-1}$. Let the known value for $B_{1:i-1}$ being identical to ${b^{\prime }}_{1:i-1}={\left[b{b^{\prime }}_1,\dots ,{b^{\prime }}_{i-1}\right]}^{\mathrm{T}}$, and define the residual interval, $r_i,$ and the random partition, $u_i,$ as in Equation (4). By the property of Dirichlet distribution and by replacing the variable $b_i$ with partition ${i^{\prime }}_i$, the PDF for $f_{B_i|B_{1:i-1}={b^{\prime }}_{1:i-1}}\left(u_i\right)$, shorten as $f_{B_i| {b^{\prime }}_{1:i-1}}\left(u_i\right)$ for brevity, could be obtained as in Equation (5). The PDF shown here indicates that $u_i$ essentially follows Beta distribution with the under-determined parameters ${\theta }_i={\alpha }_i$ and ${\varphi }_i={{\displaystyle \sum }}_{m=i}^K{\alpha }_m$. Readers can find a thorough derivation regarding this result in Appendix A.

$$r_i^{\prime }=1-{{\displaystyle \sum }}_{m=1}^{i-1}{b^{\prime }}_m , u_i=\frac{b_i}{r_i^{\prime }}$$

(4)

$$f_{B_i| {b^{\prime }}_{1:i-1}}\left(u_i\right)=\frac{\Gamma \left({\theta }_i+{\varphi }_i\right)}{\Gamma \left({\theta }_i\right)·\Gamma \left({\varphi }_i\right)}{u}_i^{{\theta }_i-1}{\left(1-u_i\right)}^{{\varphi }_i-1}$$

(5)

The under-determined parameters, ${\theta }_i$ and ${\varphi }_i$, could be determined by letting the first-order origin moment and second-order central moment of $f_{B_i| {b^{\prime }}_{1:i-1}}\left(u_i\right)$ be identical to ${\tilde{\mu }}_i={\mu }_i/r_i^{\prime }$ and ${\tilde{\sigma }}_i^2={\left({\sigma }_i/r_i^{\prime }\right)}^2$, respectively. ${\mu }_i$ and ${\sigma }_i$ are the suggested mean value and standard deviation of the decay branch ratio, $B_i$, from the evaluated nuclear data library. The solution for ${\theta }_i$ and ${\varphi }_i$ are as in Equations (6) and (7).

$${\theta }_s=\frac{{\tilde{\mu }}_s·\left({\tilde{\mu }}_s-{\tilde{\mu }}_s^2-{\tilde{\sigma }}_s^2\right)}{{\tilde{\sigma }}_s^2}$$

(6)

$${\varphi }_s=\frac{\left(1-{\tilde{\mu }}_s\right)·\left({\tilde{\mu }}_s-{\tilde{\mu }}_s^2-{\tilde{\sigma }}_s^2\right)}{{\tilde{\sigma }}_s^2}$$

(7)

From the above discussion, a Dirichlet distribution based decay branch ratio sampling method, named “DIR sampling method,” is introduced (shown in Figure 2). This sampling method is composed of two-level nested iterations: namely, an iteration over each sample (outer iteration) and an iteration over the branch ratio (inner iteration). For the $m$-th outer iteration, the $m$-th decay branch ratio vector sample, $b_{1:K}^{\left(m\right)}=\left[b_1^{\left(m\right)},b_2^{\left(m\right)},\dots ,b_K^{\left(m\right)}\right]$, will be generated iteratively in the inner iteration. There are three key steps involved in sampling each decay branch ratio sample, $b_i^{\left(m\right)},$ for any $i=1,\dots ,K-1$: “residual interval” calculation and “partition” random sampling. Specifically, the last branch ratio sample, $b_K^{\left(m\right)},$ is calculated as the residual of the preceding decay branch ratios samples, $b_1^{\left(m\right)}$ to $b_{K-1}^{\left(m\right)}$, which could impose the “sum-to-one” condition on the sampled $m$-th decay branch ratio vector sample, $b_{1:K}^{\left(m\right)}$.

It is worthwhile to note that the DIR sampling method was developed based on the conditional factorization of Dirichlet distribution and, therefore, the sampling for $b_i^{\left(m\right)}$ will be more or less dependent on the preceding sampled values, $b_{1:i-1}^{\left(m\right)}$. Therefore, the DIR sampling method could become vulnerable when the preceding sampled values, $b_{1:i-1}^{\left(m\right)}$, become too large, as this would leave no residual interval, $r_i^{\left(m\right)}=1-{{\displaystyle \sum }}_{k=1}^{i-1}{b}_k^{\left(m\right)}$, for $b_i^{\left(m\right)}$ to be sampled. This shortcoming of the DIR sampling method stems from the fact that there are less parameters for the $K-1$ variates’ Dirichlet distribution to accommodate the $2\times \left(K-1\right)$ suggested statistics (e.g., the mean and standard deviation provided in the evaluated nuclear data library) for $K-1$ branch ratios. Therefore, a more flexible sampling procedure, which is based on the generalized Dirichlet distribution, will be introduced in the following section to tackle this issue.

3.2. Development of gDIR Sampling Method

Connor and Mosimann [22] introduced the generalized Dirichlet distribution by the concept of “complete neutrality”. If the decay branch ratio random vector, $B_{1:K-1}$, follows $K-1$ variates’ generalized Dirichlet distribution, the random vector, $B_{1:K-1}$, is expected to be completely neutral and generated by a series of mutual, independent, Beta-distributed random partitions from $U_1$ to $U_{K-1}$. This mutual independency of $U_{1:K-1}$ is held by the complete neutrality of $B_{1:K-1}$.

If we assign a random partition, $U_i,$ with a Beta distribution of parameters ${\alpha }_i$ and ${\beta }_i$, denoted as $\mathrm{Beta}\left({\alpha }_i,{\beta }_i\right)$ for any $i=1,\dots ,K-1$, the generated branch ratio vector, $B_{1:K-1}$, could be derived following the generalized Dirichlet distribution with the PDF shown in Equation (8a), denoted as $\mathrm{GD}\left({\alpha }_{1:K-1};{\beta }_{1:K-1}\right)$. Here, ${\alpha }_{1:K-1}=\left({\alpha }_1,\dots ,{\alpha }_{K-1}\right)$ and ${\beta }_{1:K-1}=\left({\beta }_1,\dots ,{\beta }_{K-1}\right)$. It is noteworthy to mention that if we let ${\beta }_i={\alpha }_{i+1}+{\beta }_{i+1}$ and ${\beta }_{K-1}={\alpha }_K$, $\mathrm{GD}\left({\alpha }_{1:K-1};{\beta }_{1:K-1}\right)$ will degenerate to the Dirichlet distribution $\mathrm{Dir}\left({\alpha }_{1:K-1};{\alpha }_K\right)$ [22,23]. The relationship between Dirichlet distribution and generalized Dirichlet distribution is shown in Figure 3.

$$f_{B_{1:K-1}}\left(b_{1:K-1}\right)={\displaystyle \prod }_{i=1}^{K-1}\frac{\Gamma \left({\alpha }_i+{\beta }_i\right)}{\Gamma \left({\alpha }_i\right)\Gamma \left({\beta }_i\right)}{\displaystyle \prod }_{i=1}^{K-1}{b}_i^{{\alpha }_i-1}{\left(1-{\displaystyle \sum }_{m=1}^i{b}_m\right)}^{{\widehat{\beta }}_i}$$

(8a)

$${\widehat{\beta }}_i=\left\{\begin{array}{c}{\beta }_i-\left({\alpha }_{i+1}+{\beta }_{i+1}\right), i\le K-2\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em} {\beta }_i-1, i=K-1\end{array}\right.$$

(8b)

Given that there are $2\left(K-1\right)$ underdetermined distribution parameters in the generalized Dirichlet distribution, it provides enough “modeling” flexibility to accord with the suggested mean and standard deviation values for the uncertain decay branch ratio vector, $B_{1:K-1}$. By the general moment function of the generalized Dirichlet distribution [23,24], these underdetermined parameters can be determined as in Equation (9). Here, $i$ is the index for branch ratios, taking values from $1$ to $K-1$. Specifically, $A_0=1$ and $B_0=1$.

$${\alpha }_i=\frac{{\mu }_i^2{B}_{i-1}-{\mu }_i{A}_{i-1}\left({\sigma }_i^2+{\mu }_i^2\right)}{A_{i-1}^2\left({\sigma }_i^2+{\mu }_i^2\right)-{\mu }_i^2{B}_{i-1}}, {\beta }_i=\frac{{\alpha }_i\left(A_{i-1}-{\mu }_i\right)}{{\mu }_i}$$

(9a)

$$A_i={\displaystyle \prod }_{m=1}^i\frac{{\beta }_m}{{\alpha }_m+{\beta }_m}, B_i={\displaystyle \prod }_{m=1}^i\frac{{\beta }_m\left({\beta }_m+1\right)}{\left({\alpha }_m+{\beta }_m\right)\left({\alpha }_m+{\beta }_m+1\right)}$$

(9b)

It can be noted from Equation (10) that each $\left({\alpha }_i,{\beta }_i\right)$ is related with the corresponding mean value, ${\mu }_i$, and the standard deviation, ${\sigma }_i,$ suggested for the branch ratio, $B_i$, in the evaluated nuclear data library. Therefore, all the suggested statistics for $B_{1:K-1}$ could be strictly kept in the generalized Dirichlet distribution. A detailed sampling procedure, named after the gDIR sampling method, is presented in Figure 4.

When compared with the DIR sampling method introduced in Section 3.1, the gDIR sampling method has an identical number of underdetermined parameters with the suggested statistics of branch ratio $B_1$ to $B_{K-1}$. Therefore, the calculation of the distribution parameters is independent among these branch ratios. For example, compare the red notation in Figure 2 and Figure 4: the gDIR sampling method tackles the “vulnerability” issue that appeared in the DIR sampling method (previously mentioned at the end of Section 3.1) and becomes more robust.

Before moving on to numerical implementation in Section 4, it is worthwhile to comment that the branch ratio sampling order is crucial in implementing both the DIR and gDIR sampling methods. Specifically, given the uncertain branch ratio vector, $B_{1:K}$, different choices of the last sampled branch ratio, $B_K$ (which is calculated from the “sum-to-one” condition), would be relevant to the accuracy of its estimated standard deviation compared with its suggested value. This is mainly due to the fact that both Dirichlet and generalized Dirichlet distribution are founded upon the concept of “complete neutrality”, which is random-variate order dependent. Readers may refer to [22,23] for a detailed discussion.

4. Numerical Results and Discussion

4.1. Sampling Results from DIR and gDIR

In order to verify the performance of the DIR and gDIR sampling methods, branch ratio data of twenty radioactive isotopes were retrieved from ENDF/B-VIII.0. These data are listed in

Table 1. Branch ratio data of selected radioactive isotopes from ENDF/B-VIII.0.

Index	Radioactive Isotope	Branch Ratio (Standard Deviation) (%)
		Decay Branch I	Decay Branch II	Decay Branch III
1	Ga-79	0.0890 (0.0190)	4.9663 (0.1671)	94.9447 (0.1671)
2	Ga-81	11.900 (0.7000)	41.4771 (3.8260)	46.6229 (3.8260)
3	Rb-81M	0.0203 (0.0075)	2.3797 (0.8372)	97.6000 (0.6000)
4	Rb-98	0.0510 (0.0070)	13.8000 (0.6000)	86.1490 (0.6000)
5	Sr-97	0.0200 (0.0100)	14.8690 (1.9979)	85.1110 (1.9979)
6	Pd-111M	7.8912 (0.9316)	19.1088 (2.6269)	73.0000 (3.0000)
7	Ag-117M	6.0000 (1.5000)	20.9512 (8.8311)	73.0488 (1.5875)
8	In-127M	0.3691 (0.0869)	0.6900 (0.0400)	98.9409 (0.0869)
9	In-129	0.2500 (0.0500)	10.6205 (0.8061)	89.1295 (0.8061)
10	In-130	0.9300 (0.1300)	28.6116 (3.6308)	70.4584 (3.6308)
11	In-130N	1.6500 (0.1500)	15.0092 (2.6323)	83.3408 (2.6323)
12	Te-133M	9.9880 (0.7729)	16.5000 (2.0000)	73.5120 (4.6175)
13	Er-149M	0.1700 (0.0700)	3.5000 (0.7000)	96.3300 (0.7000)
14	Bi-196N	0.0004 (0.0001)	25.8000 (2.5000)	74.1997 (0.0025)
15	Po-201	1.6000 (0.3000)	38.9648 (2.5612)	59.4352 (3.6833)
16	Po-211M	0.0160 (0.0040)	8.9486 (0.1400)	91.0354 (0.1500)
17	Ac-226	0.0060 (0.0020)	16.9970 (3.0000)	82.9970 (3.0000)
18	Pa-230	0.0032 (0.0001)	7.7984 (0.7000)	92.1984 (0.7000)
19	Np-236	0.2000 (0.0600)	13.5000 (0.8000)	86.3000 (0.8000)
20	Am-237	0.0250 (0.0030)	28.0860 (1.1522)	71.8890 (3.1400)

. Here, only radioactive isotopes with three decay branches were chosen for this study, as a decay of more than three branches is seldom seen in the current release of the evaluated nuclear data library. Compared with the commonly seen two-branch decay, three-branch decay is difficult to sample; It is, therefore, used to justify the method in this work. A detailed description of these data is presented in Appendix B.

For each radioactive isotope, 100,000 branch ratio samples were randomly sampled for each of its three decay branches using the DIR and gDIR sampling methods. A sample scatter plot for the branch ratio of Gallium-81 (Ga-81) is shown in Figure 5. It can be seen that, because of the “sum-to-one” constraint, the variation of the branch ratio samples is restrained to a triangle surface. All the samples obtained both from DIR and gDIR lie on this surface.

Figure 6 and Figure 7 are the scatter plots of the mean values provided in ENDF/B-VIII.0 against the estimated mean values from the branch ratio samples with the DIR and gDIR sampling method, respectively. Detailed data referring to these two figures are listed in Appendix C. In these figures, there are twenty markers for each color, representing different radioactive isotopes, and there are three different colors representing decay branches I to III. It can be noted that both the DIR and gDIR sampling methods can generate branch ratio samples with estimated mean values approximating the suggested mean values in ENDF/B-VIII.0. This result indicates that both the DIR and gDIR sampling methods can generate branch ratio samples while preserving the suggested mean values provided in the evaluated nuclear data library.

Figure 8 and Figure 9 are the scatter plots of the standard deviation (STD) values provided in ENDF/B-VIII.0 against the estimated STDs from the branch ratio samples with the DIR and gDIR sampling methods, respectively. Detailed data referring to these two figures are listed in Appendix C. It can be found that, in these figures, both the DIR and gDIR sampling methods can generate branch ratio samples while preserving the suggested branch ratio STD of decay branch I and decay branch II. A few outliers are overserved in the last decay branch (decay branch III). These outliers are summarized in

Table 2. Outliers of estimated STD in DIR and gDIR sampling.

Sampling Method	Radioactive Isotope	Branch Ratio Standard Deviation (Suggested/Sampled) (%)
		I	II	III
DIR	Ag-117M	1.5000/1.4944	8.8311/8.8318	1.5875/8.9606
	Te-133M	0.7729/0.7700	2.0000/2.0000	4.6175/2.1442
	Bi-196N	0.0001/0.0001	2.5000/2.4999	0.0025/2.4999
	Am-237	0.0030/0.0030	1.1522/1.1521	3.1400/1.1522
gDIR	Ag-117M	1.5000/1.4944	8.8311/8.8303	1.5875/8.9032
	Te-133M	0.7729/0.7700	2.0000/2.0000	4.6175/2.0925
	Bi-196N	0.0001/0.0001	2.5000/2.4999	0.0025/2.4999
	Am-237	0.0030/0.0030	1.1522/1.1521	3.1400/1.1522

.

The “sum-to-one” condition is responsible for these outliers’ existence. As discussed in Section 3, given a radioactive isotope with three decay branches, both the DIR and gDIR sampling methods randomly sample branch ratios of the first two decay branches and calculate the third branches by subtracting the summation of these samples with unity. In this way, the mean value of the third decay branch ratio will be easily preserved only if the first two decay branches’ mean values are preserved. In the case of STD, the covariance data of the first two decay branches should be taken into consideration; this is not covered in both the DIR and gDIR sampling methods. Therefore, several outliers of the estimated STD are observed in branch ratio III. A more detailed discussion is given in the following Section.

4.2. Comparison and Discussion

Consider three-decay branches with uncertain decay branch ratios of $B_1$, $B_2,$ and $B_3$. Let σ_i be the standard deviation of the branch ratio, $B_i,$ suggested in the evaluated nuclear data library. $\mathrm{Take} \mathrm{cov}\left(B_i,B_j\right)$ as the covariance between B_i and $B_j$. Due to the existence of the “sum-to-one” condition, the following standard deviation for the third decay branch ratio can be found in Equation (10).

$${\sigma }_3=\sqrt{{\sigma }_1^2+{\sigma }_2^2+2\mathrm{cov}\left(B_1,B_2\right)}$$

(10)

From this relationship, it can be deduced that the if $\left|\mathrm{cov}\left(B_i,B_j\right)\right|\approx 0$, the calculated STD of branch ratio III from the “sum-to-one” condition will be preserved only if the estimated STDs for the first two branch ratios are preserved. However, it will become intractable if $\left|\mathrm{cov}\left(B_i,B_j\right)\right|>0$. In this case, the calculated STD of branch ratio III will be further affected by this covariance between the first two sampled branches, as seen in Equation (10). Here, the symbol $\left|·\right|$ represents taking the absolute value.

$${\rho }_{12}=\frac{\mathrm{Cov}\left(B_1,B_2\right)}{{\sigma }_1{\sigma }_2}=\frac{{\sigma }_3^2-{\sigma }_1^2-{\sigma }_2^2}{2{\sigma }_1{\sigma }_2}\in \left[-1,1\right]$$

(11)

The covariance data can be normalized as the correlation coefficient by the STD of the related branch rations, as in Equation (11). A detailed comparison between these correlation coefficients is presented in

Table 3. Correlation coefficient of outliers in DIR and gDIR sampling.

Sampling Method	Radioactive Isotope	Correlation Coefficient		STD of Decay Branch III
		$\mathbf{Required} {\mathit{\rho }}_1$	$\mathbf{Sampled} {\mathit{\rho }}_2$	(Suggested/Sampled) (%)
DIR	Ag-117M	−2.9	2.2 × 10−3	1.5875/8.9606
	Te-133M	5.4	1.7 × 10−3	4.6175/2.1442
	Bi-196N	1.3 × 104	1.9 × 10−3	0.0025/2.4999
	Am-237	1.2 × 103	1.6 × 10−3	3.1400/1.1522
	In-129	$-3.1 \times {10}^{-2} \approx 0$	$1.8 \times {10}^{-3} \approx 0$	0.8061/0.8077
gDIR	Ag-117M	−2.9	−3.6 × 10−2	1.5875/8.9032
	Te-133M	5.4	−6.9 × 10−2	4.6175/2.0925
	Bi-196N	1.3 × 104	1.9 × 10−3	0.0025/2.4999
	Am-237	1.2 × 103	9.1 × 10−4	3.1400/1.1522
	In-129	$-3.1 \times {10}^{-2} \approx 0$	$-4.8 \times {10}^{-3} \approx 0$	0.8061/0.8074

. Here, the “required” correlation coefficient, ${\rho }_1,$ is calculated by the STD of the branch ratios provided in ENDF/B-VIII.0, which represent the internal correlation coefficient between the first and second branch ratios. The “sampled” correlation coefficient, ${\rho }_2,$ is estimated from the sampled branch ratio I and branch ratio II samples for both the DIR and gDIR sampling methods. From this chart, in comparison with the case of isotope In-129, it can be seen that the sampled correlation coefficients deviate from the required correlation coefficients to a large degree, and these deviations cause the large bias in the estimated STD of the third decay branch. Apart from these deviations, the required correlation coefficients all exceed the bound $\left[-1,1\right]$. This indicates an inconsistency in the standard deviations of the decay branch ratio data provided by the evaluated nuclear data library [11,14]. This issue will be investigated in a further study.

5. Conclusions

In order to stochastically sample the branch ratio data given the suggested mean values and standard deviations, the present work introduced the DIR sampling method and the gDIR sampling method. These methods were developed by assuming that the branch ratios follow Dirichlet or generalized Dirichlet distribution. Branch ratio data retrieved from ENDF/B-VIII.0 were used to justify the performance of the methods. Conclusions are summarized as follows:

(1)

Both the DIR and gDIR sampling methods generate branch ratio samples which preserve their suggested mean values and standard deviations while satisfying the “sum-to-one” constraint. Due to sampling-order dependence appearing in these methods, it is recommended that the branch ratio sampling order is arranged in an ascending manner with respect to the branch ratio values;

(2)

Compared with the DIR sampling method, the gDIR sampling method is more flexible in describing the randomness of the branch ratios theoretically due to the increased number of under-determined distribution parameters. Therefore, it is expected that, in actual practices, the gDIR sampling method will be more robust;

(3)

When the branch ratios provided by the evaluated nuclear data library inherently have significant correlations among each other, both the DIR and gDIR sampling methods will fail to guarantee the preservation of the suggested standard deviation of the last sampled branch ratio.

The introduced branch-ratio sampling methods are expected to be used as alternatives in the stochastic-sampling-based uncertainty analysis of decay branch ratio data in the process of decay heat uncertainty quantification. In future work, they are expected to be used in the stochastic sampling uncertainty analysis of the spent fuel decay heat calculation and the safety analysis of HTGRs. Additionally, a detailed investigation and comparative research on the performance of Dirichlet-type sampling methods in radioactive decay branch ratio sampling problems, especially their performance in comparison with Gamma-distribution-based direct sampling procedures [25], will be conducted in the future.