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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 Thomas–Fermi model ： ウィキペディア英語版 Thomas–Fermi model The Thomas–Fermi (TF) model,〔 〕〔 〕 named after Llewellyn Thomas and Enrico Fermi, is a quantum mechanical theory for the electronic structure of many-body systems developed semiclassically shortly after the introduction of the Schrödinger equation. It stands separate from wave function theory as being formulated in terms of the electronic density alone and as such is viewed as a precursor to modern density functional theory. The TF model is correct only in the limit of an infinite nuclear charge. Using the approximation for realistic systems yields poor quantitative predictions, even failing to reproduce some general features of the density such as shell structure in atoms and Friedel oscillations in solids. It has, however, found modern applications in many fields through the ability to extract qualitative trends analytically and with the ease at which the model can be solved. The kinetic energy expression of Thomas–Fermi theory is also used as a component in more sophisticated density approximation to the kinetic energy within modern orbital-free density functional theory. Working independently, Thomas and Fermi used this statistical model in 1927 to approximate the distribution of electrons in an atom. Although electrons are distributed nonuniformly in an atom, an approximation was made that the electrons are distributed uniformly in each small volume element ''ΔV'' (i.e. locally) but the electron density $n(\backslash vec)$ can still vary from one small volume element to the next. == Kinetic energy == For a small volume element ''ΔV'', and for the atom in its ground state, we can fill out a spherical momentum space volume ''Vf''  up to the Fermi momentum ''p''''f'' , and thus,〔March 1992, p.24〕 :$V\_f\; =\; \backslash frac\backslash pi\; p\_^3(\backslash vec)$ where $\backslash vec$ is a point in ''ΔV''. The corresponding phase space volume is :$\backslash Delta\; V\_\; =\; V\_f\; \backslash \; \backslash Delta\; V\; =\; \backslash frac\backslash pi\; p\_^3(\backslash vec)\; \backslash \; \backslash Delta\; V\; .$ The electrons in ''ΔVph''  are distributed uniformly with two electrons per ''h3'' of this phase space volume, where ''h'' is Planck's constant.〔Parr and Yang 1989, p.47〕 Then the number of electrons in ''ΔVph''  is :$\backslash Delta\; N\_\; =\; \backslash frac\; \backslash \; \backslash Delta\; V\_\; =\; \backslash fracp\_^3(\backslash vec)\; \backslash \; \backslash Delta\; V\; .$ The number of electrons in ''ΔV''  is :$\backslash Delta\; N\; =\; n(\backslash vec)\; \backslash \; \backslash Delta\; V$ where $n(\backslash vec)$ is the electron density. Equating the number of electrons in ''ΔV'' to that in ''ΔVph''  gives, :$n(\backslash vec)=\backslash fracp\_^3(\backslash vec)\; .$ The fraction of electrons at $\backslash vec$ that have momentum between ''p'' and ''p+dp'' is, :$\backslash begin$ F_\vec (p) dp & = \frac \pi p_f^3(\vec)} \qquad \qquad p \le p_f(\vec) \\ & = 0 \qquad \qquad \qquad \quad \text \\ \end Using the classical expression for the kinetic energy of an electron with mass ''me'', the kinetic energy per unit volume at $\backslash vec$ for the electrons of the atom is, :$\backslash begin$ t(\vec) & = \int \frac \ n(\vec) \ F_\vec (p) \ dp \\ & = n(\vec) \int_^ \frac \ \ \frac \pi p_f^3(\vec)} \ dp \\ & = C_F \ ()^ \end where a previous expression relating $n(\backslash vec)$ to $p\_f(\backslash vec)$ has been used and, :$C\_F=\backslash frac\backslash left(\backslash frac\backslash right)^\}.$ Integrating the kinetic energy per unit volume $t(\backslash vec)$ over all space, results in the total kinetic energy of the electrons,〔March 1983, p. 5, Eq. 11〕 :$T=C\_F\backslash int\; ()^\backslash \; d^3r\; \backslash \; .$ This result shows that the total kinetic energy of the electrons can be expressed in terms of only the spatially varying electron density $n(\backslash vec)\; ,$ according to the Thomas–Fermi model. As such, they were able to calculate the energy of an atom using this expression for the kinetic energy combined with the classical expressions for the nuclear-electron and electron-electron interactions (which can both also be represented in terms of the electron density). 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「Thomas–Fermi model」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.