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Radioxenon Production and Transport from an Underground Nuclear Detonation to Ground Surface

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Abstract

Radioxenon isotopes are considered as possible indicators for detecting and discriminating underground nuclear explosions. To monitor and sample the release of radioxenon isotopes, both independent and chain-reaction yields need to be considered together with multiphase transport in geological systems from the detonation point to the ground surface. For the sake of simplicity, modeling of radioxenon isotopic radioactivities has typically been focused either on chain reactions in a batch reactor without considering multiphase transport or on radionuclide transport with simplified reactions. Although numerical methods are available for integrating coupled differential equations of complex decay networks, the stiffness of ordinary differential equations due to greatly differing decay rates may require substantial additional effort to obtain solutions for the fully coupled system. For this reason, closed-form solutions for sequential reactions and numerical solutions for multiparent converging and multidaughter branching reactions were previously developed and used to simulate xenon isotopic radioactivities in the batch reactor mode. In this paper, we develop a fully coupled numerical model, which involves tracking 24 components (i.e., 22 radionuclide components plus air and water) in two phases to enhance model predictability of simultaneously simulating xenon isotopic transport and fully coupled chain reactions. To validate the numerical model and verify the corresponding computer code, we derived closed-form solutions for first-order xenon reactions in a batch reactor mode and for single-gas phase transport coupled with the xenon reactions in a one-dimensional column. Finally, cylindrical 3-D simulations of two-phase flow within a dual permeability fracture-matrix medium, simulating the geohydrologic regime of an underground nuclear explosion, indicate the existence of both a strong temporal and spatial dependence of xenon isotopic ratios sampled at the surface. In the example presented here, temporally evolving subsurface xenon isotopic ratios are found to migrate across the discrimination line delineating civilian nuclear activities from an underground nuclear explosion in the KALINOWSKI Multi-Isotope Ratio Chart.

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Acknowledgments

The authors wish to thank Martin B. Kalinowski at Carl Friedrich von Weizsäcker Center for Science and Peace Research (ZNF) and Carol A. Velsko at Lawrence Livermore National Laboratory for providing and interpreting data of fission product decay chains. The authors also thank Nathan G. Wimer and Steven A. Kreek at Lawrence Livermore National Laboratory, two anonymous reviewers, for their careful review and helpful comments that led to an improved manuscript. This research was funded by the Office of Proliferation Detection (NA-221), U.S. Department of Energy and performed under award number DE-AC52-06NA25946 and the auspices of the U.S. Department of Energy by Lawrence Livermore National Laboratory under Contract No. DE-AC52-07NA27344. The authors also wish to express their gratitude to the National Nuclear Security Administration, Office of Defense Nuclear Nonproliferation Research and Development (DNN R&D) and the Comprehensive Inspection Technologies working group, a multi-institutional and interdisciplinary group of scientists and engineers.

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Authors and Affiliations

  1. Lawrence Livermore National Laboratory, Livermore, CA , 94550, USA

    Yunwei Sun, Charles R. Carrigan & Yue Hao

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  1. Yunwei Sun
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  2. Charles R. Carrigan
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  3. Yue Hao
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Correspondence to Yunwei Sun.

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IM release number: LLNL-JRNL-648035.

Appendices

Appendix A: Transformation Matrices of 131 and 133 Decay Networks

Transformation matrices \({\mathbf {S}}^-\) and \({\mathbf {S}}\) of 131 and 133 decay networks are formulated as:

$$\begin{aligned} {\mathbf {S}}^-=\left[ \begin{array}{cccccccc} 1 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0\\ \frac{\alpha _1k_1}{k_1-k_2} &{} 1 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0\\ S^-_{3,1} &{} \frac{\alpha _2k_2}{k_2-k_3} &{} 1 &{} 0 &{} 0 &{} 0 &{} 0 &{}0 \\ S^-_{4,1} &{} S^-_{4,2} &{} \frac{\alpha _3k_3}{k_3-k_4} &{} 1 &{} 0 &{} 0 &{} 0&{} 0\\ S^-_{5,1} &{} S^-_{5,2} &{} S^-_{5,3} &{} \frac{\alpha _6k_4}{k_4-k_5} &{} 1 &{} 0 &{} 0 &{}0 \\ S^-_{6,1} &{} S^-_{6,2} &{} S^-_{6,3} &{} S^-_{6,4} &{} \frac{k_5}{k_5-k_6} &{}1&{}0 &{}0\\ S^-_{7,1} &{} S^-_{7,2} &{} S^-_{7,3} &{} S^-_{7,4} &{} S^-_{7,5} &{} \frac{\alpha _7k_6}{k_6-k_7} &{} 1 &{} 0\\ S^-_{8,1} &{} S^-_{8,2} &{} S^-_{8,3} &{} S^-_{8,4} &{} S^-_{8,5} &{} S^-_{8,6} &{} \frac{k_7}{k_7-k_8} &{} 1\\ \end{array} \right] \end{aligned},$$
(10)

where

$$\begin{aligned} S^-_{3,1}&= \frac{\alpha _1k_1}{k_1-k_3}\cdot \frac{\alpha _2k_2}{k_2-k_3} \quad (1\,\overrightarrow{\alpha _1}\,2\,\overrightarrow{\alpha _2}\,3)\\ S^-_{4,1}&= \frac{\alpha _1k_1}{k_1-k_4}\cdot \frac{\alpha _2k_2}{k_2-k_4}\cdot \frac{\alpha _3k_3}{k_3-k_4} \quad (1\,\overrightarrow{\alpha _1}\,2\,\overrightarrow{\alpha _2}\,3\,\overrightarrow{\alpha _3}\,4)\\ S^-_{5,1}&= \frac{\alpha _1k_1}{k_1-k_5}\cdot \frac{\alpha _2k_2}{k_2-k_5}\cdot \frac{\alpha _3k_3}{k_3-k_5}\cdot \frac{\alpha _6k_4}{k_4-k_5} \quad (1\,\overrightarrow{\alpha _1}\,2\,\overrightarrow{\alpha _2}\,3\,\overrightarrow{\alpha _3}\,4\,\overrightarrow{\alpha _6}\,5)\\&+ \frac{\alpha _1k_1}{k_1-k_5}\cdot \frac{\alpha _2k_2}{k_2-k_5}\cdot \frac{\alpha _4k_3}{k_3-k_5} \quad (1\,\overrightarrow{\alpha _1}\,2\,\overrightarrow{\alpha _2}\,3\,\overrightarrow{\alpha _4}\,5)\\ S^-_{6,1}&= \frac{\alpha _1k_1}{k_1-k_6}\cdot \frac{\alpha _2k_2}{k_2-k_6}\cdot \frac{\alpha _3k_3}{k_3-k_6}\cdot \frac{\alpha _5k_4}{k_4-k_6} \quad (1\,\overrightarrow{\alpha _1}\,2\,\overrightarrow{\alpha _2}\,3\,\overrightarrow{\alpha _3}\,4\,\overrightarrow{\alpha _5}\,6)\\&+ \frac{\alpha _1k_1}{k_1-k_6}\cdot \frac{\alpha _2k_2}{k_2-k_6}\cdot \frac{\alpha _3k_3}{k_3-k_6}\cdot \frac{\alpha _6k_4}{k_4-k_6}\cdot \frac{k_5}{k_5-k_6} \quad (1\,\overrightarrow{\alpha _1}\,2\,\overrightarrow{\alpha _2}\,3\,\overrightarrow{\alpha _3}\,4\,\overrightarrow{\alpha _6}\,5 \rightarrow 6)\\&+ \frac{\alpha _1k_1}{k_1-k_6}\cdot \frac{\alpha _2k_2}{k_2-k_6}\cdot \frac{\alpha _4k_3}{k_3-k_6}\cdot \frac{k_5}{k_5-k_6} \quad (1\,\overrightarrow{\alpha _1}\,2\,\overrightarrow{\alpha _2}\,3\,\overrightarrow{\alpha _4}\,5 \rightarrow 6)\\ S^-_{7,1}&= \frac{\alpha _1k_1}{k_1-k_7}\cdot \frac{\alpha _2k_2}{k_2-k_7}\cdot \frac{\alpha _3k_3}{k_3-k_7}\cdot \frac{\alpha _5k_4}{k_4-k_7}\cdot \frac{\alpha _7k_6}{k_6-k_7} \quad (1\,\overrightarrow{\alpha _1}\,2\,\overrightarrow{\alpha _2}\,3\,\overrightarrow{\alpha _3}\,4\,\overrightarrow{\alpha _5}\,6\,\overrightarrow{\alpha _7}7)\\&+ \frac{\alpha _1k_1}{k_1-k_7}\cdot \frac{\alpha _2k_2}{k_2-k_7}\cdot \frac{\alpha _3k_3}{k_3-k_7}\cdot \frac{\alpha _6k_4}{k_4-k_7}\cdot \frac{k_5}{k_5-k_7}\cdot \frac{\alpha _7k_6}{k_6-k_7} \quad (1\,\overrightarrow{\alpha _1}\,2\,\overrightarrow{\alpha _2}\,3\,\overrightarrow{\alpha _3}\,4\,\overrightarrow{\alpha _6}\,5 \rightarrow 6\,\overrightarrow{\alpha _7}\,7)\\&+ \frac{\alpha _1k_1}{k_1-k_7}\cdot \frac{\alpha _2k_2}{k_2-k_7}\cdot \frac{\alpha _4k_3}{k_3-k_7}\cdot \frac{k_5}{k_5-k_7}\cdot \frac{\alpha _7k_6}{k_6-k_7} \quad (1\,\overrightarrow{\alpha _1}\,2\,\overrightarrow{\alpha _2}\,3\,\overrightarrow{\alpha _4}\,5 \rightarrow 6\,\overrightarrow{\alpha _7}\,7)\\ S^-_{8,1}&= \frac{\alpha _1k_1}{k_1-k_8}\cdot \frac{\alpha _2k_2}{k_2-k_8}\cdot \frac{\alpha _3k_3}{k_3-k_8}\cdot \frac{\alpha _5k_4}{k_4-k_8}\cdot \frac{\alpha _7k_6}{k_6-k_8}\cdot \frac{k_7}{k_7-k_8} \quad (1\,\overrightarrow{\alpha _1}\,2\,\overrightarrow{\alpha _2}\,3\,\overrightarrow{\alpha _3}\,4\,\overrightarrow{\alpha _5}\,6\,\overrightarrow{\alpha _7}\,7\rightarrow 8)\\&+ \frac{\alpha _1k_1}{k_1-k_8}\cdot \frac{\alpha _2k_2}{k_2-k_8}\cdot \frac{\alpha _3k_3}{k_3-k_8}\cdot \frac{\alpha _5k_4}{k_4-k_8}\cdot \frac{\alpha _8k_6}{k_6-k_8} \quad (1\,\overrightarrow{\alpha _1}\,2\,\overrightarrow{\alpha _2}\,3\,\overrightarrow{\alpha _3}\,4\,\overrightarrow{\alpha _5}\,6\,\overrightarrow{\alpha _8}\,8)\\&+ \frac{\alpha _1k_1}{k_1-k_8}\cdot \frac{\alpha _2k_2}{k_2-k_8}\cdot \frac{\alpha _3k_3}{k_3-k_8}\cdot \frac{\alpha _6k_4}{k_4-k_8}\cdot \frac{k_5}{k_5-k_8}\cdot \frac{\alpha _7k_6}{k_6-k_8}\cdot \frac{k_7}{k_7-k_8} \quad (1\,\overrightarrow{\alpha _1}\,2\,\overrightarrow{\alpha _2}\,3\,\overrightarrow{\alpha _3}\,4\,\overrightarrow{\alpha _6}\,5 \rightarrow 6\,\overrightarrow{\alpha _7}\,7\rightarrow 8)\\&+ \frac{\alpha _1k_1}{k_1-k_8}\cdot \frac{\alpha _2k_2}{k_2-k_8}\cdot \frac{\alpha _3k_3}{k_3-k_8}\cdot \frac{\alpha _6k_4}{k_4-k_8}\cdot \frac{k_5}{k_5-k_8}\cdot \frac{\alpha _8k_6}{k_6-k_8} \quad (1\,\overrightarrow{\alpha _1}\,2\,\overrightarrow{\alpha _2}\,3\,\overrightarrow{\alpha _3}\,4\,\overrightarrow{\alpha _6}\,5 \rightarrow 6\,\overrightarrow{\alpha _8}\,8)\\&+ \frac{\alpha _1k_1}{k_1-k_8}\cdot \frac{\alpha _2k_2}{k_2-k_8}\cdot \frac{\alpha _4k_3}{k_3-k_8}\cdot \frac{k_5}{k_5-k_8}\cdot \frac{\alpha _7k_6}{k_6-k_8}\cdot \frac{k_7}{k_7-k_8} \quad (1\,\overrightarrow{\alpha _1}\,2\,\overrightarrow{\alpha _2}\,3\,\overrightarrow{\alpha _4}\,5 \rightarrow 6\,\overrightarrow{\alpha _7}\,7\rightarrow 8)\\&+ \frac{\alpha _1k_1}{k_1-k_8}\cdot \frac{\alpha _2k_2}{k_2-k_8}\cdot \frac{\alpha _4k_3}{k_3-k_8}\cdot \frac{k_5}{k_5-k_8}\cdot \frac{\alpha _8k_6}{k_6-k_8} \quad (1\,\overrightarrow{\alpha _1}\,2\,\overrightarrow{\alpha _2}\,3\,\overrightarrow{\alpha _4}\,5 \rightarrow 6\,\overrightarrow{\alpha _8}\,8)\\ S^-_{4,2}&= \frac{\alpha _2k_2}{k_2-k_4}\cdot \frac{\alpha _3k_3}{k_3-k_4} \quad (2\,\overrightarrow{\alpha _2}\,3\,\overrightarrow{\alpha _3}\,4)\\ S^-_{5,2}&= \frac{\alpha _2k_2}{k_2-k_5}\cdot \frac{\alpha _3k_3}{k_3-k_5}\cdot \frac{\alpha _6k_4}{k_4-k_5} \quad (2\,\overrightarrow{\alpha _2}\,3\,\overrightarrow{\alpha _3}\,4\,\overrightarrow{\alpha _6}\,5)\\&+ \frac{\alpha _2k_2}{k_2-k_5}\cdot \frac{\alpha _4k_3}{k_3-k_5} \quad (2\,\overrightarrow{\alpha _2}\,3\,\overrightarrow{\alpha _4}\,5)\\ S^-_{6,2}&= \frac{\alpha _2k_2}{k_2-k_6}\cdot \frac{\alpha _3k_3}{k_3-k_6}\cdot \frac{\alpha _5k_4}{k_4-k_6} \quad (2\,\overrightarrow{\alpha _2}\,3\,\overrightarrow{\alpha _3}\,4\,\overrightarrow{\alpha _5}\,6)\\&+ \frac{\alpha _2k_2}{k_2-k_6}\cdot \frac{\alpha _3k_3}{k_3-k_6}\cdot \frac{\alpha _6k_4}{k_4-k_6}\cdot \frac{k_5}{k_5-k_6} \quad (2\,\overrightarrow{\alpha _2}\,3\,\overrightarrow{\alpha _3}\,4\,\overrightarrow{\alpha _6}\,5\rightarrow 6)\\&+ \frac{\alpha _2k_2}{k_2-k_6}\cdot \frac{\alpha _4k_3}{k_3-k_6}\cdot \frac{k_5}{k_5-k_6} \quad (2\,\overrightarrow{\alpha _2}\,3\,\overrightarrow{\alpha _4}\,5\rightarrow 6)\\ S^-_{7,2}&= \frac{\alpha _2k_2}{k_2-k_7}\cdot \frac{\alpha _3k_3}{k_3-k_7}\cdot \frac{\alpha _5k_4}{k_4-k_7}\cdot \frac{\alpha _7k_6}{k_6-k_7} \quad (2\,\overrightarrow{\alpha _2}\,3\,\overrightarrow{\alpha _3}\,4\,\overrightarrow{\alpha _5}\,6\,\overrightarrow{\alpha _7}\,7)\\&+ \frac{\alpha _2k_2}{k_2-k_7}\cdot \frac{\alpha _3k_3}{k_3-k_7}\cdot \frac{\alpha _6k_4}{k_4-k_7}\cdot \frac{k_5}{k_5-k_7}\cdot \frac{\alpha _7k_6}{k_6-k_7} \quad (2\,\overrightarrow{\alpha _2}\,3\,\overrightarrow{\alpha _3}\,4\,\overrightarrow{\alpha _6}\,5 \rightarrow 6\,\overrightarrow{\alpha _7}\,7)\\&+ \frac{\alpha _2k_2}{k_2-k_7}\cdot \frac{\alpha _4k_3}{k_3-k_7}\cdot \frac{k_5}{k_5-k_7}\cdot \frac{\alpha _7k_6}{k_6-k_7} \quad (2\,\overrightarrow{\alpha _2}\,3\,\overrightarrow{\alpha _4}\,5 \rightarrow 6\,\overrightarrow{\alpha _7}\,7)\\ S^-_{8,2}&= \frac{\alpha _2k_2}{k_2-k_8}\cdot \frac{\alpha _3k_3}{k_3-k_8}\cdot \frac{\alpha _5k_4}{k_4-k_8}\cdot \frac{\alpha _7k_6}{k_6-k_8}\cdot \frac{k_7}{k_7-k_8} \quad (2\,\overrightarrow{\alpha _2}\,3\,\overrightarrow{\alpha _3}\,4\,\overrightarrow{\alpha _5}\,6\,\overrightarrow{\alpha _7}\,7\rightarrow 8)\\&+ \frac{\alpha _2k_2}{k_2-k_8}\cdot \frac{\alpha _3k_3}{k_3-k_8}\cdot \frac{\alpha _5k_4}{k_4-k_8}\cdot \frac{\alpha _8k_6}{k_6-k_8} \quad (2\,\overrightarrow{\alpha _2}\,3\,\overrightarrow{\alpha _3}\,4\,\overrightarrow{\alpha _5}\,6\,\overrightarrow{\alpha _8}\,8)\\&+ \frac{\alpha _2k_2}{k_2-k_8}\cdot \frac{\alpha _3k_3}{k_3-k_8}\cdot \frac{\alpha _6k_4}{k_4-k_8}\cdot \frac{k_5}{k_5-k_8}\cdot \frac{\alpha _7k_6}{k_6-k_8}\cdot \frac{k_7}{k_7-k_8} \quad (2\,\overrightarrow{\alpha _2}\,3\,\overrightarrow{\alpha _3}\,4\,\overrightarrow{\alpha _6}\,5 \rightarrow 6\,\overrightarrow{\alpha _7}\,7\rightarrow 8)\\&+ \frac{\alpha _2k_2}{k_2-k_8}\cdot \frac{\alpha _3k_3}{k_3-k_8}\cdot \frac{\alpha _6k_4}{k_4-k_8}\cdot \frac{k_5}{k_5-k_8}\cdot \frac{\alpha _8k_6}{k_6-k_8} \quad (2\,\overrightarrow{\alpha _2}\,3\,\overrightarrow{\alpha _3}\,4\,\overrightarrow{\alpha _6}\,5 \rightarrow 6\,\overrightarrow{\alpha _8}\,8)\\&+ \frac{\alpha _2k_2}{k_2-k_8}\cdot \frac{\alpha _4k_3}{k_3-k_8}\cdot \frac{k_5}{k_5-k_8}\cdot \frac{\alpha _7k_6}{k_6-k_8}\cdot \frac{k_7}{k_7-k_8} \quad (2\,\overrightarrow{\alpha _2}\,3\,\overrightarrow{\alpha _4}\,5 \rightarrow 6\,\overrightarrow{\alpha _7}\,7\rightarrow 8)\\&+ \frac{\alpha _2k_2}{k_2-k_8}\cdot \frac{\alpha _4k_3}{k_3-k_8}\cdot \frac{k_5}{k_5-k_8}\cdot \frac{\alpha _8k_6}{k_6-k_8} \quad (2\,\overrightarrow{\alpha _2}\,3\,\overrightarrow{\alpha _4}\,5 \rightarrow 6\,\overrightarrow{\alpha _8}\,8)\\ S^-_{5,3}&= \frac{\alpha _3k_3}{k_3-k_5}\cdot \frac{\alpha _6k_4}{k_4-k_5} \quad (3\,\overrightarrow{\alpha _3}\,4\,\overrightarrow{\alpha _6}\,5)\\&+ \frac{\alpha _4k_3}{k_3-k_5} \quad (3\,\overrightarrow{\alpha _4}\,5)\\ S^-_{6,3}&= \frac{\alpha _3k_3}{k_3-k_6}\cdot \frac{\alpha _5k_4}{k_4-k_6} \quad (3\,\overrightarrow{\alpha _3}\,4\,\overrightarrow{\alpha _5}\,6)\\&+ \frac{\alpha _3k_3}{k_3-k_6}\cdot \frac{\alpha _6k_4}{k_4-k_6}\cdot \frac{k_5}{k_5-k_6} \quad (3\,\overrightarrow{\alpha _3}\,4\,\overrightarrow{\alpha _6}\,5\rightarrow 6)\\&+ \frac{\alpha _4k_3}{k_3-k_6}\cdot \frac{k_5}{k_5-k_6} \quad (3\,\overrightarrow{\alpha _4}\,5\rightarrow 6)\\ S^-_{7,3}&= \frac{\alpha _3k_3}{k_3-k_7}\cdot \frac{\alpha _5k_4}{k_4-k_7}\cdot \frac{\alpha _7k_6}{k_6-k_7} \quad (3\,\overrightarrow{\alpha _3}\,4\,\overrightarrow{\alpha _5}\,6\,\overrightarrow{\alpha _7}\,7)\\&+ \frac{\alpha _3k_3}{k_3-k_7}\cdot \frac{\alpha _6k_4}{k_4-k_7}\cdot \frac{k_5}{k_5-k_7}\cdot \frac{\alpha _7k_6}{k_6-k_7} \quad (3\,\overrightarrow{\alpha _3}\,4\,\overrightarrow{\alpha _6}\,5\,\rightarrow 6\,\overrightarrow{\alpha _7}\,7)\\&+ \frac{\alpha _4k_3}{k_3-k_7}\cdot \frac{k_5}{k_5-k_7}\cdot \frac{\alpha _7k_6}{k_6-k_7} \quad (3\,\overrightarrow{\alpha _4}\,5 \rightarrow 6\,\overrightarrow{\alpha _7}\,7)\\ S^-_{8,3}&= \frac{\alpha _3k_3}{k_3-k_8}\cdot \frac{\alpha _5k_4}{k_4-k_8}\cdot \frac{\alpha _7k_6}{k_6-k_8}\cdot \frac{k_7}{k_7-k_8} \quad (3\,\overrightarrow{\alpha _3}\,4\,\overrightarrow{\alpha _5}\,6\,\overrightarrow{\alpha _7}\,7\rightarrow 8)\\&+ \frac{\alpha _3k_3}{k_3-k_8}\cdot \frac{\alpha _5k_4}{k_4-k_8}\cdot \frac{\alpha _8k_6}{k_6-k_8} \quad (3\,\overrightarrow{\alpha _3}\,4\,\overrightarrow{\alpha _5}\,6\,\overrightarrow{\alpha _8}\,8)\\&+ \frac{\alpha _3k_3}{k_3-k_8}\cdot \frac{\alpha _6k_4}{k_4-k_8}\cdot \frac{k_5}{k_5-k_8}\cdot \frac{\alpha _7k_6}{k_6-k_8}\cdot \frac{k_7}{k_7-k_8} \quad (3\,\overrightarrow{\alpha _3}\,4\,\overrightarrow{\alpha _6}\,5 \rightarrow 6\,\overrightarrow{\alpha _7}\,7\rightarrow 8)\\&+ \frac{\alpha _3k_3}{k_3-k_8}\cdot \frac{\alpha _6k_4}{k_4-k_8}\cdot \frac{k_5}{k_5-k_8}\cdot \frac{\alpha _8k_6}{k_6-k_8} \quad (3\,\overrightarrow{\alpha _3}\,4\,\overrightarrow{\alpha _6}\,5 \rightarrow 6\,\overrightarrow{\alpha _8}\,8)\\&+ \frac{\alpha _4k_3}{k_3-k_8}\cdot \frac{k_5}{k_5-k_8}\cdot \frac{\alpha _7k_6}{k_6-k_8}\cdot \frac{k_7}{k_7-k_8} \quad (3\,\overrightarrow{\alpha _4}\,5 \rightarrow 6\,\overrightarrow{\alpha _7}\,7\rightarrow 8)\\&+ \frac{\alpha _4k_3}{k_3-k_8}\cdot \frac{k_5}{k_5-k_8}\cdot \frac{\alpha _8k_6}{k_6-k_8} \quad (3\,\overrightarrow{\alpha _4}\,5 \rightarrow 6\,\overrightarrow{\alpha _8}\,8)\\ S^-_{6,4}&= \frac{\alpha _5k_4}{k_4-k_6} \quad (4\,\overrightarrow{\alpha _5}\,6)\\&+ \frac{\alpha _6k_4}{k_4-k_6}\cdot \frac{k_5}{k_5-k_6} \quad (4\,\overrightarrow{\alpha _6}\,5\rightarrow 6)\\ S^-_{7,4}&= \frac{\alpha _5k_4}{k_4-k_7}\cdot \frac{\alpha _7k_6}{k_6-k_7} \quad (4\,\overrightarrow{\alpha _5}\,6\,\overrightarrow{\alpha _7}\,7)\\&+ \frac{\alpha _6k_4}{k_4-k_7}\cdot \frac{k_5}{k_5-k_7}\cdot \frac{\alpha _7k_6}{k_6-k_7} \quad (4\,\overrightarrow{\alpha _6}\,5\rightarrow 6\,\overrightarrow{\alpha _7}\,7)\\ S^-_{8,4}&= \frac{\alpha _5k_4}{k_4-k_8}\cdot \frac{\alpha _7k_6}{k_6-k_8}\cdot \frac{k_7}{k_7-k_8} \quad (4\,\overrightarrow{\alpha _5}\,6\,\overrightarrow{\alpha _7}\,7\rightarrow 8)\\&+ \frac{\alpha _5k_4}{k_4-k_8}\cdot \frac{\alpha _8k_6}{k_6-k_8} \quad (4\,\overrightarrow{\alpha _5}\,6\,\overrightarrow{\alpha _8}\,8)\\&+ \frac{\alpha _6k_4}{k_4-k_8}\cdot \frac{k_5}{k_5-k_8}\cdot \frac{\alpha _7k_6}{k_6-k_8}\cdot \frac{k_7}{k_7-k_8} \quad (4\,\overrightarrow{\alpha _6}\,5\,\rightarrow 6\,\overrightarrow{\alpha _7}\,7\rightarrow 8)\\&+ \frac{\alpha _6k_4}{k_4-k_8}\cdot \frac{k_5}{k_5-k_8}\cdot \frac{\alpha _8k_6}{k_6-k_8} \quad (4\,\overrightarrow{\alpha _6}\,5\,\rightarrow 6\,\overrightarrow{\alpha _8}\,8) \\ S^-_{7,5}&= \frac{k_5}{k_5-k_7}\cdot \frac{\alpha _7k_6}{k_6-k_7} \quad (5\rightarrow 6\,\overrightarrow{\alpha _7}\,7) \\ S^-_{8,5}&= \frac{k_5}{k_5-k_8}\cdot \frac{\alpha _7k_6}{k_6-k_8}\cdot \frac{k_7}{k_7-k_8} \quad (5\rightarrow 6\,\overrightarrow{\alpha _7}\,7\rightarrow 8) \\&+ \frac{k_5}{k_5-k_8}\cdot \frac{\alpha _8k_6}{k_6-k_8} \quad (5\rightarrow 6\,\overrightarrow{\alpha _8}\,8)\\ S^-_{8,6}&= \frac{\alpha _7k_6}{k_6-k_8}\cdot \frac{k_7}{k_7-k_8} \quad (6\,\overrightarrow{\alpha _7}\,7\rightarrow 8)\\&+ \frac{\alpha _8k_6}{k_6-k_8} \quad (6\,\overrightarrow{\alpha _8}\,8), \end{aligned}$$

and

$$\begin{aligned} {\mathbf {S}}=\left[ \begin{array}{cccccccc} 1 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{}0\\ \frac{\alpha _1k_1}{k_2-k_1} &{} 1 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{}0\\ S_{3,1} &{} \frac{\alpha _2k_2}{k_3-k_2} &{} 1 &{} 0 &{} 0 &{} 0 &{} 0 &{}0\\ S_{4,1} &{} S_{4,2} &{} \frac{\alpha _3k_3}{k_4-k_3} &{} 1 &{} 0 &{} 0 &{} 0 &{}0\\ S_{5,1} &{} S_{5,2} &{} S_{5,3} &{} \frac{\alpha _6k_4}{k_5-k_4} &{} 1 &{} 0 &{} 0 &{}0\\ S_{6,1} &{} S_{6,2} &{} S_{6,3} &{} S_{6,4} &{} \frac{k_5}{k_6-k_5} &{} 1 &{} 0 &{}0\\ S_{7,1} &{} S_{7,2} &{} S_{7,3} &{} S_{7,4} &{} S_{7,5} &{} \frac{\alpha _7k_6}{k_7-k_6} &{} 1 &{}0\\ S_{8,1} &{} S_{8,2} &{} S_{8,3} &{} S_{8,4} &{} S_{8,5} &{} S_{8,6} &{} \frac{k_7}{k_8-k_7} &{}1\\ \end{array} \right] \end{aligned},$$
(11)

where

$$\begin{aligned} S_{3,1}&= \frac{\alpha _1k_1}{k_3-k_1}\cdot \frac{\alpha _2k_2}{k_2-k_1} \quad (1\,\overrightarrow{\alpha _1}\,2\,\overrightarrow{\alpha _2}\,3)\\ S_{4,1}&= \frac{\alpha _1k_1}{k_4-k_1}\cdot \frac{\alpha _2k_2}{k_2-k_1}\cdot \frac{\alpha _3k_3}{k_3-k_1} \quad (1\,\overrightarrow{\alpha _1}\,2\,\overrightarrow{\alpha _2}\,3\,\overrightarrow{\alpha _3}\,4)\\ S_{5,1}&= \frac{\alpha _1k_1}{k_5-k_1}\cdot \frac{\alpha _2k_2}{k_2-k_1}\cdot \frac{\alpha _3k_3}{k_3-k_1}\cdot \frac{\alpha _6k_4}{k_4-k_1} \quad (1\,\overrightarrow{\alpha _1}\,2\,\overrightarrow{\alpha _2}\,3\,\overrightarrow{\alpha _3}\,4\,\overrightarrow{\alpha _6}\,5)\\&\quad + \frac{\alpha _1k_1}{k_5-k_1}\cdot \frac{\alpha _2k_2}{k_2-k_1}\cdot \frac{\alpha _4k_3}{k_3-k_1} \quad (1\,\overrightarrow{\alpha _1}\,2\,\overrightarrow{\alpha _2}\,3\,\overrightarrow{\alpha _4}\,4)\\ S_{6,1}&= \frac{\alpha _1k_1}{k_6-k_1}\cdot \frac{\alpha _2k_2}{k_2-k_1}\cdot \frac{\alpha _3k_3}{k_3-k_1}\cdot \frac{\alpha _6k_4}{k_4-k_1}\cdot \frac{k_5}{k_5-k_1} \quad (1\,\overrightarrow{\alpha _1}\,2\,\overrightarrow{\alpha _2}\,3\,\overrightarrow{\alpha _3}\,4\,\overrightarrow{\alpha _6}\,5\rightarrow 6)\\&+ \frac{\alpha _1k_1}{k_6-k_1}\cdot \frac{\alpha _2k_2}{k_2-k_1}\cdot \frac{\alpha _3k_3}{k_3-k_1}\cdot \frac{\alpha _5k_4}{k_4-k_1} \quad (1\,\overrightarrow{\alpha _1}\,2\,\overrightarrow{\alpha _2}\,3\,\overrightarrow{\alpha _3}\,4\,\overrightarrow{\alpha _5}\,6)\\&+ \frac{\alpha _1k_1}{k_6-k_1}\cdot \frac{\alpha _2k_2}{k_2-k_1}\cdot \frac{\alpha _4k_3}{k_3-k_1}\cdot \frac{k_5}{k_5-k_1} \quad (1\,\overrightarrow{\alpha _1}\,2\,\overrightarrow{\alpha _2}\,3\,\overrightarrow{\alpha _4}\,5\rightarrow 6)\\ S_{7,1}&= \frac{\alpha _1k_1}{k_7-k_1}\cdot \frac{\alpha _2k_2}{k_2-k_1}\cdot \frac{\alpha _3k_3}{k_3-k_1}\cdot \frac{\alpha _5k_4}{k_4-k_1}\cdot \frac{\alpha _7k_6}{k_6-k_1} \quad (1\,\overrightarrow{\alpha _1}\,2\,\overrightarrow{\alpha _2}\,3\,\overrightarrow{\alpha _5}\,6\,\overrightarrow{\alpha _7}\,7)\\&+ \frac{\alpha _1k_1}{k_7-k_1}\cdot \frac{\alpha _2k_2}{k_2-k_1}\cdot \frac{\alpha _3k_3}{k_3-k_1}\cdot \frac{\alpha _6k_4}{k_4-k_1}\cdot \frac{k_5}{k_5-k_1}\cdot \frac{\alpha _7k_6}{k_6-k_1} \quad (1\,\overrightarrow{\alpha _1}\,2\,\overrightarrow{\alpha _2}\,3\,\overrightarrow{\alpha _3}\,4\,\overrightarrow{\alpha _6}\,5\rightarrow 6\,\overrightarrow{\alpha _7}\,7)\\&+ \frac{\alpha _1k_1}{k_7-k_1}\cdot \frac{\alpha _2k_2}{k_2-k_1}\cdot \frac{\alpha _4k_3}{k_3-k_1}\cdot \frac{k_5}{k_5-k_1}\cdot \frac{\alpha _7k_6}{k_6-k_1} \quad (1\,\overrightarrow{\alpha _1}\,2\,\overrightarrow{\alpha _2}\,3\,\overrightarrow{\alpha _4}\,5 \rightarrow 6\,\overrightarrow{\alpha _7}\,7)\\ S_{8,1}&= \frac{\alpha _1k_1}{k_8-k_1}\cdot \frac{\alpha _2k_2}{k_2-k_1}\cdot \frac{\alpha _3k_3}{k_3-k_1}\cdot \frac{\alpha _5k_4}{k_4-k_1}\cdot \frac{\alpha _7k_6}{k_6-k_1}\cdot \frac{k_7}{k_7-k_1} \quad (1\,\overrightarrow{\alpha _1}\,2\,\overrightarrow{\alpha _2}\,3\,\overrightarrow{\alpha _3}\,4\,\overrightarrow{\alpha _5}\,6\,\overrightarrow{\alpha _7}\,7\rightarrow 8)\\&+ \frac{\alpha _1k_1}{k_8-k_1}\cdot \frac{\alpha _2k_2}{k_2-k_1}\cdot \frac{\alpha _3k_3}{k_3-k_1}\cdot \frac{\alpha _5k_4}{k_4-k_1}\cdot \frac{\alpha _8k_6}{k_6-k_1} \quad (1\,\overrightarrow{\alpha _1}\,2\,\overrightarrow{\alpha _2}\,3\,\overrightarrow{\alpha _3}\,4\,\overrightarrow{\alpha _5}\,6\,\overrightarrow{\alpha _8}\,8)\\&+ \frac{\alpha _1k_1}{k_8-k_1}\cdot \frac{\alpha _2k_2}{k_2-k_1}\cdot \frac{\alpha _3k_3}{k_3-k_1}\cdot \frac{\alpha _6k_4}{k_4-k_1}\cdot \frac{k_5}{k_5-k_1}\cdot \frac{\alpha _7k_6}{k_6-k_1}\cdot \frac{k_7}{k_7-k_1} \quad (1\,\overrightarrow{\alpha _1}\,2\,\overrightarrow{\alpha _2}\,3\,\overrightarrow{\alpha _3}\,4\,\overrightarrow{\alpha _6}\,5\,\rightarrow 6\,\overrightarrow{\alpha _7}\,7\rightarrow 8)\\&+ \frac{\alpha _1k_1}{k_8-k_1}\cdot \frac{\alpha _2k_2}{k_2-k_1}\cdot \frac{\alpha _3k_3}{k_3-k_1}\cdot \frac{\alpha _6k_4}{k_4-k_1}\cdot \frac{k_5}{k_5-k_1}\cdot \frac{\alpha _8k_6}{k_6-k_1} \quad (1\,\overrightarrow{\alpha _1}\,2\,\overrightarrow{\alpha _3}\,3\,\overrightarrow{\alpha _3}\,4\,\overrightarrow{\alpha _6}\,5\rightarrow 6\,\overrightarrow{\alpha _8}\,8)\\&+ \frac{\alpha _1k_1}{k_8-k_1}\cdot \frac{\alpha _2k_2}{k_2-k_1}\cdot \frac{\alpha _4k_3}{k_3-k_1}\cdot \frac{k_5}{k_5-k_1}\cdot \frac{\alpha _7k_6}{k_6-k_1}\cdot \frac{k_7}{k_7-k_1} \quad (1\,\overrightarrow{\alpha _1}\,2\,\overrightarrow{\alpha _2}\,3\,\overrightarrow{\alpha _4}\,5 \rightarrow 6\,\overrightarrow{\alpha _7}\,7\rightarrow 8)\\&+ \frac{\alpha _1k_1}{k_8-k_1}\cdot \frac{\alpha _2k_2}{k_2-k_1}\cdot \frac{\alpha _4k_3}{k_3-k_1}\cdot \frac{k_5}{k_5-k_1}\cdot \frac{\alpha _8k_6}{k_6-k_1} \quad (1\,\overrightarrow{\alpha _1}\,2\,\overrightarrow{\alpha _2}\,3\,\overrightarrow{\alpha _4}\,5 \rightarrow 6\,\overrightarrow{\alpha _8}\,8)\\ S_{4,2}&= \frac{\alpha _2k_2}{k_4-k_2}\cdot \frac{\alpha _3k_3}{k_3-k_2} \quad (2\,\overrightarrow{\alpha _2}\,3\,\overrightarrow{\alpha _3}\,4)\\ S_{5,2}&= \frac{\alpha _2k_2}{k_5-k_2}\cdot \frac{\alpha _3k_3}{k_3-k_2}\cdot \frac{\alpha _6k_4}{k_4-k_2} \quad (2\,\overrightarrow{\alpha _2}\,3\,\overrightarrow{\alpha _3}\,4\,\overrightarrow{\alpha _6}\,5)\\&+ \frac{\alpha _2k_2}{k_5-k_2}\cdot \frac{\alpha _4k_3}{k_3-k_2} \quad (2\,\overrightarrow{\alpha _2}\,3\,\overrightarrow{\alpha _4}\,5)\\ S_{6,2}&= \frac{\alpha _2k_2}{k_6-k_2}\cdot \frac{\alpha _3k_3}{k_3-k_2}\cdot \frac{\alpha _5k_4}{k_4-k_2} \quad (2\,\overrightarrow{\alpha _2}\,3\,\overrightarrow{\alpha _3}\,4\,\overrightarrow{\alpha _5}\,6)\\&+ \frac{\alpha _2k_2}{k_6-k_2}\cdot \frac{\alpha _3k_3}{k_3-k_2}\cdot \frac{\alpha _6k_4}{k_4-k_2}\cdot \frac{k_5}{k_5-k_2} \quad (2\,\overrightarrow{\alpha _2}\,3\,\overrightarrow{\alpha _3}\,4\,\overrightarrow{\alpha _6}\,5\rightarrow 6)\\&+ \frac{\alpha _2k_2}{k_6-k_2}\cdot \frac{\alpha _4k_3}{k_3-k_2}\cdot \frac{k_5}{k_5-k_2} \quad (2\,\overrightarrow{\alpha _2}\,3\,\overrightarrow{\alpha _4}\,5\rightarrow 6)\\ S_{7,2}&= \frac{\alpha _2k_2}{k_7-k_2}\cdot \frac{\alpha _3k_3}{k_3-k_2}\cdot \frac{\alpha _5k_4}{k_4-k_2}\cdot \frac{\alpha _7k_6}{k_6-k_2} \quad (2\,\overrightarrow{\alpha _2}\,3\,\overrightarrow{\alpha _3}\,4\,\overrightarrow{\alpha _5}\,6\,\overrightarrow{\alpha _7}\,7)\\&+ \frac{\alpha _2k_2}{k_7-k_2}\cdot \frac{\alpha _3k_3}{k_3-k_2}\cdot \frac{\alpha _6k_4}{k_4-k_2}\cdot \frac{k_5}{k_5-k_2}\cdot \frac{\alpha _7k_6}{k_6-k_2} \quad (2\,\overrightarrow{\alpha _2}\,3\,\overrightarrow{\alpha _3}\,4\,\overrightarrow{\alpha _6}\,5 \rightarrow 6\,\overrightarrow{\alpha _7}\,7)\\&+ \frac{\alpha _2k_2}{k_7-k_2}\cdot \frac{\alpha _4k_3}{k_3-k_2}\cdot \frac{k_5}{k_5-k_2}\cdot \frac{\alpha _7k_6}{k_6-k_2} \quad (2\,\overrightarrow{\alpha _2}\,3\,\overrightarrow{\alpha _4}\,5 \rightarrow 6\,\overrightarrow{\alpha _7}\,7)\\ S_{8,2}&= \frac{\alpha _2k_2}{k_8-k_2}\cdot \frac{\alpha _3k_3}{k_3-k_2}\cdot \frac{\alpha _5k_4}{k_4-k_2}\cdot \frac{\alpha _7k_6}{k_6-k_2}\cdot \frac{k_7}{k_7-k_2} \quad (2\,\overrightarrow{\alpha _2}\,3\,\overrightarrow{\alpha _3}\,4\,\overrightarrow{\alpha _5}\,6\,\overrightarrow{\alpha _7}\,7\rightarrow 8)\\&+ \frac{\alpha _2k_2}{k_8-k_2}\cdot \frac{\alpha _3k_3}{k_3-k_2}\cdot \frac{\alpha _5k_4}{k_4-k_2}\cdot \frac{\alpha _8k_6}{k_6-k_2} \quad (2\,\overrightarrow{\alpha _2}\,3\,\overrightarrow{\alpha _3}\,4\,\overrightarrow{\alpha _5}\,6\,\overrightarrow{\alpha _8}\,8)\\&+ \frac{\alpha _2k_2}{k_8-k_2}\cdot \frac{\alpha _3k_3}{k_3-k_2}\cdot \frac{\alpha _6k_4}{k_4-k_2}\cdot \frac{k_5}{k_5-k_2}\cdot \frac{\alpha _7k_6}{k_6-k_2}\cdot \frac{k_7}{k_7-k_2} \quad (2\,\overrightarrow{\alpha _2}\,3\,\overrightarrow{\alpha _3}\,4\,\overrightarrow{\alpha _6}\,5 \rightarrow 6\,\overrightarrow{\alpha _7}\,7\rightarrow 8)\\&+ \frac{\alpha _2k_2}{k_8-k_2}\cdot \frac{\alpha _3k_3}{k_3-k_2}\cdot \frac{\alpha _6k_4}{k_4-k_2}\cdot \frac{k_5}{k_5-k_2}\cdot \frac{\alpha _8k_6}{k_6-k_2} \quad (2\,\overrightarrow{\alpha _2}\,3\,\overrightarrow{\alpha _3}\,4\,\overrightarrow{\alpha _6}\,5 \rightarrow 6\,\overrightarrow{\alpha _8}\,7)\\&+ \frac{\alpha _2k_2}{k_8-k_2}\cdot \frac{\alpha _4k_3}{k_3-k_2}\cdot \frac{k_5}{k_5-k_2}\cdot \frac{\alpha _7k_6}{k_6-k_2}\cdot \frac{k_7}{k_7-k_2} \quad (2\,\overrightarrow{\alpha _2}\,3\,\overrightarrow{\alpha _4}\,5 \rightarrow 6\,\overrightarrow{\alpha _7}\,7\rightarrow 8)\\&+ \frac{\alpha _2k_2}{k_8-k_2}\cdot \frac{\alpha _4k_3}{k_3-k_2}\cdot \frac{k_5}{k_5-k_2}\cdot \frac{\alpha _8k_6}{k_6-k_2} \quad (2\,\overrightarrow{\alpha _2}\,3\,\overrightarrow{\alpha _4}\,5 \rightarrow 6\,\overrightarrow{\alpha _8}\,8)\\ S_{5,3}&= \frac{\alpha _3k_3}{k_5-k_3}\cdot \frac{\alpha _6k_4}{k_4-k_3} \quad (3\,\overrightarrow{\alpha _3}\,4\,\overrightarrow{\alpha _6}\,5)\\&+ \frac{\alpha _4k_3}{k_5-k_3} \quad (3\,\overrightarrow{\alpha _4}\,5)\\ S_{6,3}&= \frac{\alpha _3k_3}{k_6-k_3}\cdot \frac{\alpha _5k_4}{k_4-k_3} \quad (3\,\overrightarrow{\alpha _3}\,4\,\overrightarrow{\alpha _5}\,6)\\&+ \frac{\alpha _3k_3}{k_6-k_3}\cdot \frac{\alpha _6k_4}{k_4-k_3}\cdot \frac{k_5}{k_5-k_3} \quad (3\,\overrightarrow{\alpha _3}\,4\,\overrightarrow{\alpha _6}\,5\rightarrow 6)\\&+ \frac{\alpha _4k_3}{k_6-k_3}\cdot \frac{k_5}{k_5-k_3} \quad (3\,\overrightarrow{\alpha _4}\,5\rightarrow 6)\\ S_{7,3}&= \frac{\alpha _3k_3}{k_7-k_3}\cdot \frac{\alpha _5k_4}{k_4-k_3}\cdot \frac{\alpha _7k_6}{k_6-k_3} \quad (3\,\overrightarrow{\alpha _3}\,4\,\overrightarrow{\alpha _5}\,6\,\overrightarrow{\alpha _7}\,7)\\&+ \frac{\alpha _3k_3}{k_7-k_3}\cdot \frac{\alpha _6k_4}{k_4-k_3}\cdot \frac{k_5}{k_5-k_3}\cdot \frac{\alpha _7k_6}{k_6-k_3} \quad (3\,\overrightarrow{\alpha _3}\,4\,\overrightarrow{\alpha _6}\,5\,\rightarrow 6\,\overrightarrow{\alpha _7}\,7)\\&+ \frac{\alpha _4k_3}{k_7-k_3}\cdot \frac{k_5}{k_5-k_3}\cdot \frac{\alpha _7k_6}{k_6-k_3} \quad (3\,\overrightarrow{\alpha _4}\,5 \rightarrow 6\,\overrightarrow{\alpha _7}\,7)\\ S_{8,3}&= \frac{\alpha _3k_3}{k_8-k_3}\cdot \frac{\alpha _5k_4}{k_4-k_3}\cdot \frac{\alpha _7k_6}{k_6-k_3}\cdot \frac{k_7}{k_7-k_3} \quad (3\,\overrightarrow{\alpha _3}\,4\,\overrightarrow{\alpha _5}\,6\,\overrightarrow{\alpha _7}\,7\rightarrow 8)\\&+ \frac{\alpha _3k_3}{k_8-k_3}\cdot \frac{\alpha _5k_4}{k_4-k_3}\cdot \frac{\alpha _8k_6}{k_6-k_3} \quad (3\,\overrightarrow{\alpha _3}\,4\,\overrightarrow{\alpha _5}\,6\,\overrightarrow{\alpha _8}\,8)\\&+ \frac{\alpha _3k_3}{k_8-k_3}\cdot \frac{\alpha _6k_4}{k_4-k_3}\cdot \frac{k_5}{k_5-k_3}\cdot \frac{\alpha _7k_6}{k_6-k_3}\cdot \frac{k_7}{k_7-k_3} \quad (3\,\overrightarrow{\alpha _3}\,4\,\overrightarrow{\alpha _6}\,5 \rightarrow 6\,\overrightarrow{\alpha _7}\,7\rightarrow 8)\\&+ \frac{\alpha _3k_3}{k_8-k_3}\cdot \frac{\alpha _6k_4}{k_4-k_3}\cdot \frac{k_5}{k_5-k_3}\cdot \frac{\alpha _8k_6}{k_6-k_3} \quad (3\,\overrightarrow{\alpha _3}\,4\,\overrightarrow{\alpha _6}\,5 \rightarrow 6\,\overrightarrow{\alpha _8}\,8)\\&+ \frac{\alpha _4k_3}{k_8-k_3}\cdot \frac{k_5}{k_5-k_3}\cdot \frac{\alpha _7k_6}{k_6-k_3}\cdot \frac{k_7}{k_7-k_3} \quad (3\,\overrightarrow{\alpha _4}\,5 \rightarrow 6\,\overrightarrow{\alpha _7}\,7\rightarrow 8)\\&+ \frac{\alpha _4k_3}{k_8-k_3}\cdot \frac{k_5}{k_5-k_3}\cdot \frac{\alpha _8k_6}{k_6-k_3} \quad (3\,\overrightarrow{\alpha _4}\,5 \rightarrow 6\,\overrightarrow{\alpha _8}\,8)\\ S_{6,4}&= \frac{\alpha _5k_4}{k_6-k_4} \quad (4\,\overrightarrow{\alpha _5}\,6)\\&+ \frac{\alpha _6k_4}{k_6-k_4}\cdot \frac{k_5}{k_5-k_4} \quad (4\,\overrightarrow{\alpha _6}\,5\rightarrow 6)\\ S_{7,4}&= \frac{\alpha _5k_4}{k_7-k_4}\cdot \frac{\alpha _7k_6}{k_6-k_4} \quad (4\,\overrightarrow{\alpha _5}\,6\,\overrightarrow{\alpha _7}\,7)\\&+ \frac{\alpha _6k_4}{k_7-k_4}\cdot \frac{k_5}{k_5-k_4}\cdot \frac{\alpha _7k_6}{k_6-k_4} \quad (4\,\overrightarrow{\alpha _6}\,5\rightarrow 6\,\overrightarrow{\alpha _7}\,7)\\ S_{8,4}&= \frac{\alpha _5k_4}{k_8-k_4}\cdot \frac{\alpha _7k_6}{k_6-k_4}\cdot \frac{k_7}{k_7-k_4} \quad (4\,\overrightarrow{\alpha _5}\,6\,\overrightarrow{\alpha _7}\,7\rightarrow 8)\\&+ \frac{\alpha _5k_4}{k_8-k_4}\cdot \frac{\alpha _8k_6}{k_6-k_4} \quad (4\,\overrightarrow{\alpha _5}\,6\,\overrightarrow{\alpha _8}\,8)\\&+ \frac{\alpha _6k_4}{k_8-k_4}\cdot \frac{k_5}{k_5-k_4}\cdot \frac{\alpha _7k_6}{k_6-k_4}\cdot \frac{k_7}{k_7-k_4} \quad (4\,\overrightarrow{\alpha _6}\,5\,\rightarrow 6\,\overrightarrow{\alpha _7}\,7\rightarrow 8)\\&+ \frac{\alpha _6k_4}{k_8-k_4}\cdot \frac{k_5}{k_5-k_4}\cdot \frac{\alpha _8k_6}{k_6-k_4} \quad (4\,\overrightarrow{\alpha _6}\,5\,\rightarrow 6\,\overrightarrow{\alpha _8}\,8) \\ S_{7,5}&= \frac{k_5}{k_7-k_5}\cdot \frac{\alpha _7k_6}{k_6-k_5} \quad (5\rightarrow 6\,\overrightarrow{\alpha _7}\,7) \\ S_{8,5}&= \frac{k_5}{k_8-k_5}\cdot \frac{\alpha _7k_6}{k_6-k_5}\cdot \frac{k_7}{k_7-k_5} \quad (5\rightarrow 6\,\overrightarrow{\alpha _7}\,7\rightarrow 8) \\&+ \frac{k_5}{k_8-k_5}\cdot \frac{\alpha _8k_6}{k_6-k_5} \quad (5\rightarrow 6\,\overrightarrow{\alpha _8}\,8)\\ S_{8,6}&= \frac{\alpha _7k_6}{k_8-k_6}\cdot \frac{k_7}{k_7-k_6} \quad (6\,\overrightarrow{\alpha _7}\,7\rightarrow 8)\\&+ \frac{\alpha _8k_6}{k_8-k_6} \quad (6\,\overrightarrow{\alpha _8}\,8). \end{aligned}$$

Appendix B: Transformation Matrices of 135 Decay Network

Transformation matrices \({\mathbf {S}}^-\) and \({\mathbf {S}}\) of 135 decay networks are formulated as:

$$\begin{aligned} {\mathbf {S}}^-=\left[ \begin{array}{cccccc} 1 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0\\ \frac{k_1}{k_1-k_2} &{} 1 &{} 0 &{} 0 &{} 0 &{} 0\\ S^-_{3,1} &{} \frac{\alpha _1k_2}{k_2-k_3} &{} 1 &{} 0 &{} 0 &{} 0 \\ S^-_{4,1} &{} S^-_{4,2} &{} \frac{k_3}{k_3-k_4} &{} 1 &{} 0 &{} 0\\ S^-_{5,1} &{} S^-_{5,2} &{} S^-_{5,3} &{} \frac{\alpha _2k_4}{k_4-k_5} &{} 1 &{} 0\\ S^-_{6,1} &{} S^-_{6,2} &{} S^-_{6,3} &{} S^-_{6,4} &{} \frac{k_5}{k_5-k_6} &{} 1\\ \end{array} \right] \end{aligned},$$
(12)

where

$$\begin{aligned} S^-_{3,1}&= \frac{k_1}{k_1-k_3}\cdot \frac{\alpha _1k_2}{k_2-k_3} \quad (1\rightarrow 2\,\overrightarrow{\alpha _1}\,3)\\ S^-_{4,1}&= \frac{k_1}{k_1-k_4}\cdot \frac{\alpha _1k_2}{k_2-k_4}\cdot \frac{k_3}{k_3-k_4} \quad (1\rightarrow 2\,\overrightarrow{\alpha _1}\,2\rightarrow 3\rightarrow 4)\\ S^-_{5,1}&= \frac{k_1}{k_1-k_5}\cdot \frac{\alpha _1k_2}{k_2-k_5}\cdot \frac{k_3}{k_3-k_5}\cdot \frac{\alpha _2k_4}{k_4-k_5} \quad (1\rightarrow 2\,\overrightarrow{\alpha _1}\,2\rightarrow 3\rightarrow 4\,\overrightarrow{\alpha _2}\,5)\\ S^-_{6,1}&= \frac{k_1}{k_1-k_6}\cdot \frac{k_2}{k_2-k_6}\cdot \frac{\alpha _1\cdot k_3}{k_3-k_6}\cdot \frac{\alpha _3k_4}{k_4-k_6} \quad (1\rightarrow 2\,\overrightarrow{\alpha _1}\,2\rightarrow 3\rightarrow 4\,\overrightarrow{\alpha _3}\,6)\\&+ \frac{k1}{k_1-k_6}\cdot \frac{k_2}{k_2-k_6}\cdot \frac{\alpha _1k_3}{k_3-k_6}\cdot \frac{\alpha _2k_4}{k_4-k_6}\cdot \frac{k_5}{k_5-k_6} \quad (1\rightarrow 2\,\overrightarrow{\alpha _1}\,2\rightarrow 3\rightarrow 4\,\overrightarrow{\alpha _2}\,5\rightarrow 6)\\ S^-_{4,2}&= \frac{\alpha _1k_2}{k_2-k_4}\cdot \frac{k_3}{k_3-k_4} \quad (2\,\overrightarrow{\alpha _1}\,3\rightarrow 4)\\ S^-_{5,2}&= \frac{\alpha _1k_2}{k_2-k_5}\cdot \frac{k_3}{k_k3-k_5}\cdot \frac{\alpha _2k_4}{k_4-k_5} \quad (2\,\overrightarrow{\alpha _1}\,3\,\rightarrow 4\,\overrightarrow{\alpha _2}\,5)\\ S^-_{6,2}&= \frac{k_2}{k_2-k_6}\cdot \frac{\alpha _1\cdot k_3}{k_3-k_6}\cdot \frac{\alpha _3k_4}{k_4-k_6} \quad (2\,\overrightarrow{\alpha _1}\,2\rightarrow 3\rightarrow 4\,\overrightarrow{\alpha _3}\,6)\\&+ \cdot \frac{k_2}{k_2-k_6}\cdot \frac{\alpha _1k_3}{k_3-k_6}\cdot \frac{\alpha _2k_4}{k_4-k_6}\cdot \frac{k_5}{k_5-k_6} \quad (2\,\overrightarrow{\alpha _1}\,3\rightarrow 4\,\overrightarrow{\alpha _2}\,5\rightarrow 6)\\ S^-_{5,3}&= \frac{k_3}{k_3-k_5}\cdot \frac{\alpha _2k_4}{k_4-k_5} \quad (3\rightarrow 4\,\overrightarrow{\alpha _2}\, 5)\\ S^-_{6,3}&= \frac{\alpha _1\cdot k_3}{k_3-k_6}\cdot \frac{\alpha _3k_4}{k_4-k_6} \quad (3\rightarrow 4\,\overrightarrow{\alpha _3}\,6)\\&+ \frac{\alpha _1k_3}{k_3-k_6}\cdot \frac{\alpha _2k_4}{k_4-k_6}\cdot \frac{k_5}{k_5-k_6} \quad (3\rightarrow 4\,\overrightarrow{\alpha _2}\,5\rightarrow 6)\\ S^-_{6,4}&= \frac{\alpha _3k_4}{k_4-k_6} \quad (4\,\overrightarrow{\alpha _3}\,6)\\&+ \frac{\alpha _2k_4}{k_4-k_6}\cdot \frac{k_5}{k_5-k_6} \quad (4\,\overrightarrow{\alpha _2}\,5\rightarrow 6)\\ \end{aligned}$$

and

$$\begin{aligned} {\mathbf {S}}=\left[ \begin{array}{cccccc} 1 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0\\ \frac{k_1}{k_2-k_1} &{} 1 &{} 0 &{} 0 &{} 0 &{} 0\\ S_{3,1} &{} \frac{\alpha _1k_2}{k_3-k_2} &{} 1 &{} 0 &{} 0 &{} 0 \\ S_{4,1} &{} S_{4,2} &{} \frac{k_3}{k_4-k_3} &{} 1 &{} 0 &{} 0\\ S_{5,1} &{} S_{5,2} &{} S_{5,3} &{} \frac{\alpha _2k_4}{k_5-k_4} &{} 1 &{} 0\\ S_{6,1} &{} S_{6,2} &{} S_{6,3} &{} S_{6,4} &{} \frac{k_5}{k_6-k_3} &{} 1\\ \end{array} \right] \end{aligned},$$
(13)

where

$$\begin{aligned} S_{3,1}&= \frac{k_1}{k_3-k_1}\cdot \frac{\alpha _1k_2}{k_2-k_1} \quad (1\rightarrow 2\,\overrightarrow{\alpha _1}\,3)\\ S_{4,1}&= \frac{k_1}{k_4-k_1}\cdot \frac{\alpha _1k_2}{k_2-k_1}\cdot \frac{k_3}{k_3-k_1} \quad (1\rightarrow 2\,\overrightarrow{\alpha _1}\,2\rightarrow 3\rightarrow 4)\\ S_{5,1}&= \frac{k_1}{k_5-k_1}\cdot \frac{\alpha _1k_2}{k_2-k_1}\cdot \frac{k_3}{k_3-k_1}\cdot \frac{\alpha _2k_4}{k_4-k_1} \quad (1\rightarrow 2\,\overrightarrow{\alpha _1}\,2\rightarrow 3\rightarrow 4\,\overrightarrow{\alpha _2}\,5)\\ S_{6,1}&= \frac{k_1}{k_6-k_1}\cdot \frac{k_2}{k_2-k_1}\cdot \frac{\alpha _1\cdot k_3}{k_3-k_1}\cdot \frac{\alpha _3k_4}{k_4-k_1} \quad (1\rightarrow 2\,\overrightarrow{\alpha _1}\,2\rightarrow 3\rightarrow 4\,\overrightarrow{\alpha _3}\,6)\\&+ \frac{k1}{k_6-k_1}\cdot \frac{k_2}{k_2-k_1}\cdot \frac{\alpha _1k_3}{k_3-k_1}\cdot \frac{\alpha _2k_4}{k_4-k_1}\cdot \frac{k_5}{k_5-k_1} \quad (1\rightarrow 2\,\overrightarrow{\alpha _1}\,2\rightarrow 3\rightarrow 4\,\overrightarrow{\alpha _2}\,5\rightarrow 6)\\ S_{4,2}&= \frac{\alpha _1k_2}{k_4-k_2}\cdot \frac{k_3}{k_3-k_2} \quad (2\,\overrightarrow{\alpha _1}\,3\rightarrow 4)\\ S_{5,2}&= \frac{\alpha _1k_2}{k_5-k_2}\cdot \frac{k_3}{k_k3-k_2}\cdot \frac{\alpha _2k_4}{k_4-k_2} \quad (2\,\overrightarrow{\alpha _1}\,3\,\rightarrow 4\,\overrightarrow{\alpha _2}\,5)\\ S_{6,2}&= \frac{k_2}{k_6-k_2}\cdot \frac{\alpha _1\cdot k_3}{k_3-k_2}\cdot \frac{\alpha _3k_4}{k_4-k_2} \quad (2\,\overrightarrow{\alpha _1}\,2\rightarrow 3\rightarrow 4\,\overrightarrow{\alpha _3}\,6)\\&+ \cdot \frac{k_2}{k_2-k_6}\cdot \frac{\alpha _1k_3}{k_3-k_6}\cdot \frac{\alpha _2k_4}{k_4-k_6}\cdot \frac{k_5}{k_5-k_6} \quad (2\,\overrightarrow{\alpha _1}\,3\rightarrow 4\,\overrightarrow{\alpha _2}\,5\rightarrow 6)\\ S_{5,3}&= \frac{k_3}{k_5-k_3}\cdot \frac{\alpha _2k_4}{k_4-k_3} \quad (3\rightarrow 4\,\overrightarrow{\alpha _2}\, 5)\\ S_{6,3}&= \frac{\alpha _1\cdot k_3}{k_6-k_3}\cdot \frac{\alpha _3k_4}{k_4-k_3} \quad (3\rightarrow 4\,\overrightarrow{\alpha _3}\,6)\\&+ \frac{\alpha _1k_3}{k_6-k_3}\cdot \frac{\alpha _2k_4}{k_4-k_3}\cdot \frac{k_5}{k_5-k_3} \quad (3\rightarrow 4\,\overrightarrow{\alpha _2}\,5\rightarrow 6)\\ S_{6,4}&= \frac{\alpha _3k_4}{k_6-k_4} \quad (4\,\overrightarrow{\alpha _3}\,6)\\&+ \frac{\alpha _2k_4}{k_6-k_4}\cdot \frac{k_5}{k_5-k_4} \quad (4\,\overrightarrow{\alpha _2}\,5\rightarrow 6). \end{aligned}.$$

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Sun, Y., Carrigan, C.R. & Hao, Y. Radioxenon Production and Transport from an Underground Nuclear Detonation to Ground Surface. Pure Appl. Geophys. 172, 243–265 (2015). https://doi.org/10.1007/s00024-014-0863-2

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