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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 Debye sphere ： ウィキペディア英語版 Debye length In plasmas and electrolytes the Debye length (also called Debye radius), named after the Dutch physicist and physical chemist Peter Debye, is the measure of a charge carrier's net electrostatic effect in solution, and how far those electrostatic effects persist. A Debye sphere is a volume whose radius is the Debye length, in which there is a sphere of influence, and outside of which charges are electrically screened. The notion of Debye length plays an important role in plasma physics, electrolytes and colloids (DLVO theory). == Physical origin == The Debye length arises naturally in the thermodynamic description of large systems of mobile charges. In a system of $N$ different species of charges, the $j$-th species carries charge $q\_j$ and has concentration $n\_j(\backslash mathbf)$ at position $\backslash mathbf$. According to the so-called "primitive model", these charges are distributed in a continuous medium that is characterized only by its relative static permittivity, $\backslash varepsilon\_r$. This distribution of charges within this medium gives rise to an electric potential $\backslash Phi(\backslash mathbf)$ that satisfies Poisson's equation: :$\backslash varepsilon\; \backslash nabla^2\; \backslash Phi(\backslash mathbf)\; =\; -\backslash ,\; \backslash sum\_^N\; q\_j\; \backslash ,\; n\_j(\backslash mathbf)\; -\; \backslash rho\_E(\backslash mathbf)$, where $\backslash varepsilon\; \backslash equiv\; \backslash varepsilon\_r\backslash varepsilon\_0$, $\backslash varepsilon\_0$ is the electric constant, and $\backslash rho\_E$ is a charge density external (logically, not spatially) to the medium. The mobile charges not only establish $\backslash Phi(\backslash mathbf)$ but also move in response to the associated Coulomb force, $-\; q\_j\; \backslash ,\; \backslash nabla\; \backslash Phi(\backslash mathbf)$. If we further assume the system to be in thermodynamic equilibrium with a heat bath at absolute temperature $T$, then the concentrations of discrete charges, $n\_j(\backslash mathbf)$, may be considered to be thermodynamic (ensemble) averages and the associated electric potential to be a thermodynamic mean field. With these assumptions, the concentration of the $j$-th charge species is described by the Boltzmann distribution, :$n\_j(\backslash mathbf)\; =\; n\_j^0\; \backslash ,\; \backslash exp\backslash left(\; -\; \backslash frac\; \backslash right)$, where $k\_B$ is Boltzmann's constant and where $n\_j^0$ is the mean concentration of charges of species $j$. Identifying the instantaneous concentrations and potential in the Poisson equation with their mean-field counterparts in Boltzmann's distribution yields the Poisson–Boltzmann equation: :$\backslash varepsilon\; \backslash nabla^2\; \backslash Phi(\backslash mathbf)\; =\; -\backslash ,\; \backslash sum\_^N\; q\_j\; n\_j^0\; \backslash ,\; \backslash exp\backslash left(-\; \backslash frac\; \backslash right)\; -\; \backslash rho\_E(\backslash mathbf)$. Solutions to this nonlinear equation are known for some simple systems. Solutions for more general systems may be obtained in the high-temperature (weak coupling) limit, $q\_j\; \backslash ,\; \backslash Phi(\backslash mathbf)\; \backslash ll\; k\_B\; T$, by Taylor expanding the exponential: :$\backslash exp\backslash left(-\; \backslash frac\; \backslash right)\; \backslash approx$ 1 - \frac. This approximation yields the linearized Poisson-Boltzmann equation :$\backslash varepsilon\; \backslash nabla^2\; \backslash Phi(\backslash mathbf)\; =$ \left(\sum_^N \frac \right)\, \Phi(\mathbf) -\, \sum_^N n_j^0 q_j - \rho_E(\mathbf) which also is known as the Debye–Hückel equation:〔See 〕 The second term on the right-hand side vanishes for systems that are electrically neutral. The term in parentheses divided by $\backslash varepsilon$, has the units of an inverse length squared and by dimensional analysis leads to the definition of the characteristic length scale :$\backslash lambda\_D\; =$ \left(\frac\right)^ that commonly is referred to as the Debye–Hückel length. As the only characteristic length scale in the Debye–Hückel equation, $\backslash lambda\_D$ sets the scale for variations in the potential and in the concentrations of charged species. All charged species contribute to the Debye–Hückel length in the same way, regardless of the sign of their charges. For an electrically neutral system, the Poisson equation becomes :$\backslash nabla^2\; \backslash Phi(\backslash mathbf)\; =$ \lambda_D^ \Phi(\mathbf) - \frac To illustrate Debye screening, the potential produced by an external point charge $\backslash rho\_E=Q\backslash delta(\backslash mathbf)$ is :$\backslash Phi(\backslash mathbf)\; =\; \backslash frace^$ The bare Coulomb potential is exponentially screened by the medium, over a distance of the Debye length. The Debye–Hückel length may be expressed in terms of the Bjerrum length $\backslash lambda\_B$ as :$\backslash lambda\_D\; =$ \left(4 \pi \, \lambda_B \, \sum_^N n_j^0 \, z_j^2\right)^, where $z\_j\; =\; q\_j/e$ is the integer charge number that relates the charge on the $j$-th ionic species to the elementary charge $e$. 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「Debye length」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.