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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 Momentum-transfer cross section ： ウィキペディア英語版 Momentum-transfer cross section In physics, and especially scattering theory, the momentum-transfer cross section (sometimes known as the momentum-''transport'' cross section) is an effective scattering cross section useful for describing the average momentum transferred from a particle when it collides with a target. Essentially, it contains all the information about a scattering process necessary for calculating average momentum transfers but ignores other details about the scattering angle. The momentum-transfer cross section $\backslash sigma\_\; \backslash sigma\}\; (\backslash theta)$ by :$\backslash sigma\_\; \backslash sigma\}\; (\backslash theta)\; \backslash mathrm\; \backslash Omega$ :::$=\; \backslash int\; \backslash int\; (1\; -\; \backslash cos\; \backslash theta)\; \backslash frac\; (\backslash theta)\; \backslash sin\; \backslash theta\; \backslash mathrm\; \backslash theta\; \backslash mathrm\; \backslash phi$. The momentum-transfer cross section can be written in terms of the phase shifts from a partial wave analysis as :$\backslash sigma\_\; \backslash sum\_^\backslash infty\; (l+1)\; \backslash sin^2(-\; \backslash delta\_l(k)).$ == Explanation == The factor of $1\; -\; \backslash cos\; \backslash theta$ arises as follows. Let the incoming particle be traveling along the $z$-axis with vector momentum :$\backslash vec\_\backslash mathrm\; =\; q\; \backslash hat$. Suppose the particle scatters off the target with polar angle $\backslash theta$ and azimuthal angle $\backslash phi$ plane. Its new momentum is :$\backslash vec\_\backslash mathrm\; =\; q\; \backslash cos\; \backslash theta\; \backslash hat\; +\; q\; \backslash sin\; \backslash theta\; \backslash cos\; \backslash phi\backslash hat\; +\; q\; \backslash sin\; \backslash theta\; \backslash cos\; \backslash phi\backslash hat$. By conservation of momentum, the target has acquired momentum :$\backslash Delta\; \backslash vec\; =\; \backslash vec\_\backslash mathrm\; -\; \backslash vec\_\backslash mathrm\; =\; q\; (1\; -\; \backslash cos\; \backslash theta)\; \backslash hat\; -\; q\; \backslash sin\; \backslash theta\; \backslash cos\; \backslash phi\backslash hat\; -\; q\; \backslash sin\; \backslash theta\; \backslash cos\; \backslash phi\backslash hat$. Now, if many particles scatter off the target, and the target is assumed to have azimuthal symmetry, then the radial ($x$ and $y$) components of the transferred momentum will average to zero. The average momentum transfer will be just $q\; (1\; -\; \backslash cos\; \backslash theta)\; \backslash hat$. If we do the full averaging over all possible scattering events, we get :$\backslash Delta\; \backslash vec\_\backslash mathrm\; =\; \backslash langle\; \backslash Delta\; \backslash vec\; \backslash rangle\_\backslash Omega$. ::::$=\; \backslash sigma\_\backslash mathrm^\; \backslash int\; \backslash Delta\; \backslash vec\; \backslash frac\; (\backslash theta)\; \backslash mathrm\; \backslash Omega$. ::::$=\; \backslash sigma\_\backslash mathrm^\; \backslash int\; \backslash left(q\; (1\; -\; \backslash cos\; \backslash theta)\; \backslash hat\; -\; q\; \backslash sin\; \backslash theta\; \backslash cos\; \backslash phi\backslash hat\; -\; q\; \backslash sin\; \backslash theta\; \backslash cos\; \backslash phi\backslash hat\; \backslash right)\; \backslash frac\; (\backslash theta)\; \backslash mathrm\; \backslash Omega$ ::::$=\; q\; \backslash hat\; \backslash sigma\_\backslash mathrm^\; \backslash int\; (1\; -\; \backslash cos\; \backslash theta)\; \backslash frac\; (\backslash theta)\; \backslash mathrm\; \backslash Omega$ ::::$=\; q\; \backslash hat\; \backslash sigma\_\backslash mathrm\; /\; \backslash sigma\_\backslash mathrm$ where the total cross section is :$\backslash sigma\_\backslash mathrm\; =\; \backslash int\; \backslash frac\; (\backslash theta)\; \backslash mathrm\; \backslash Omega$. Therefore, for a given total cross section, one does not need to compute new integrals for every possible momentum in order to determine the average momentum transferred to a target. One just needs to compute $\backslash sigma\_\backslash mathrm$. 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「Momentum-transfer cross section」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.