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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(\mathbf) at position \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, \varepsilon_r.
This distribution of charges within this medium gives rise to an electric potential \Phi(\mathbf) that satisfies Poisson's equation:
: \varepsilon \nabla^2 \Phi(\mathbf) = -\, \sum_^N q_j \, n_j(\mathbf) - \rho_E(\mathbf),
where \varepsilon \equiv \varepsilon_r\varepsilon_0, \varepsilon_0 is the electric constant, and \rho_E is a charge density external (logically, not spatially) to the medium.
The mobile charges not only establish \Phi(\mathbf) but also move in response to the associated Coulomb force, - q_j \, \nabla \Phi(\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(\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(\mathbf) = n_j^0 \, \exp\left( - \frac \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:
: \varepsilon \nabla^2 \Phi(\mathbf) = -\, \sum_^N q_j n_j^0 \, \exp\left(- \frac \right) - \rho_E(\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 \, \Phi(\mathbf) \ll k_B T, by Taylor expanding the exponential:
: \exp\left(- \frac \right) \approx
1 - \frac.
This approximation yields the linearized Poisson-Boltzmann equation
: \varepsilon \nabla^2 \Phi(\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 \varepsilon, has the units of an inverse length squared and by
dimensional analysis leads to the definition of the characteristic length scale
: \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, \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
: \nabla^2 \Phi(\mathbf) =
\lambda_D^ \Phi(\mathbf) - \frac

To illustrate Debye screening, the potential produced by an external point charge \rho_E=Q\delta(\mathbf) is
: \Phi(\mathbf) = \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 \lambda_B as
: \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)
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