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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 Sommerfeld expansion ： ウィキペディア英語版 Sommerfeld expansion A Sommerfeld expansion is an approximation method developed by Arnold Sommerfeld for a certain class of integrals which are common in condensed matter and statistical physics. Physically, the integrals represent statistical averages using the Fermi–Dirac distribution. When the inverse temperature $\backslash beta$ is a large quantity, the integral can be expanded〔.〕〔http://www.physik.uni-regensburg.de/forschung/fabian/pages/mainframes/teaching/teaching_files/files%20of%20mf_statistical_physics/Sommerfeld.pdf〕 in terms of $\backslash beta$ as :$\backslash int\_^\backslash infty\; \backslash frac\backslash ,\backslash mathrm\backslash varepsilon\; =\; \backslash int\_^\backslash mu\; H(\backslash varepsilon)\backslash ,\backslash mathrm\backslash varepsilon\; +\; \backslash frac\backslash left(\backslash frac\backslash right)^2H^\backslash prime(\backslash mu)\; +\; O\; \backslash left(\backslash frac\backslash right)^4$ where $H^\backslash prime(\backslash mu)$ is used to denote the derivative of $H(\backslash varepsilon)$ evaluated at $\backslash varepsilon\; =\; \backslash mu$ and where the $O(x^n)$ notation refers to limiting behavior of order $x^n$. The expansion is only valid if $H(\backslash varepsilon)$ vanishes as $\backslash varepsilon\; \backslash rightarrow\; -\backslash infty$ and goes no faster than polynomially in $\backslash varepsilon$ as $\backslash varepsilon\; \backslash rightarrow\; \backslash infty$. == Application to the free electron model == Integrals of this type appear frequently when calculating electronic properties in the free electron model of solids. In these calculations the above integral expresses the expected value of the quantity $H(\backslash varepsilon)$. For these integrals we can then identify $\backslash beta$ as the inverse temperature and $\backslash mu$ as the chemical potential. Therefore, the Sommerfeld expansion is valid for large $\backslash beta$ (low temperature) systems. 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「Sommerfeld expansion」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.