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翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Bateman Equation ： ウィキペディア英語版 Bateman Equation In nuclear physics, the Bateman equation is a mathematical model describing abundances and activities in a decay chain as a function of time, based on the decay rates and initial abundances.〔H. Bateman. "Solution of a System of Differential Equations Occurring in the Theory of Radio-active Transformations," Proc. Cambridge Phil. Soc. IS, 423 (1910) https://archive.org/details/cbarchive_122715_solutionofasystemofdifferentia1843 〕 If, at time t, there are $N\_i(t)$ atoms of isotope $i$ that decays into isotope $i+1$ at the rate $\backslash lambda\_i$, the amounts of isotopes in the k-step decay chain evolves as: :$\backslash frac=-\backslash lambda\_1\; N\_1(t)$ :$\backslash frac=-\backslash lambda\_i\; N\_i(t)\; +\; \backslash lambda\_N\_(t)$ :$\backslash frac=\; \backslash lambda\_N\_(t)$ (this can be adapted to handle decay branches). While this can be solved explicitly for i=2, the formulas quickly become cumbersome for longer chains.〔http://chemistry.sfu.ca/assets/uploads/file/Course%20Materials%2012-1/NUSC%20342/L9.pdf〕 Bateman found a general explicit formula for the amounts by taking the Laplace transform of the variables. ::$N\_n(t)\; =\; \backslash sum\_^n\; \backslash left\; (N\_i(0)\; \backslash times\; \backslash left\; (\; \backslash prod\_^\; \backslash lambda\_j\; \backslash right\; )\; \backslash times$ \left ( \sum_^n \left ( \frac^n (\lambda_p-\lambda_j)} \right ) \right ) \right ) (it can also be expanded with source terms, if more atoms of isotope i are provided externally at a constant rate).〔http://www.nucleonica.com/wiki/index.php?title=Help:Decay_Engine〕 While the Bateman formula can be implemented easily in computer code, if $\backslash lambda\_p\; \backslash approx\; \backslash lambda\_j$ for some isotope pair, cancellation can lead to computational errors. Therefore, other methods such as numerical integration or the matrix exponential method are also in use.〔Logan J. Harr. Precise Calculation of Complex Radioactive Decay Chains. M.Sc thesis Air Force Institute of Technology. 2007. http://www.dtic.mil/dtic/tr/fulltext/u2/a469273.pdf〕 ==See also== * Harry Bateman * List of equations in nuclear and particle physics * Transient equilibrium * Secular equilibrium 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Bateman Equation」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.