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| In [[plasma (physics)|plasma]]s 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 [[Electric-field screening|electrically screened]]. The notion of Debye length plays an important role in [[plasma physics]], [[electrolytes]] and [[colloids]] ([[DLVO theory]]).
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| ==Physical origin==
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| The Debye length arises naturally in the thermodynamic description of large systems of mobile charges. In a system of <math>N</math> different species of charges, the <math>j</math>-th species carries charge <math>q_j</math> and has [[concentration]] <math>n_j(\mathbf{r})</math> at position <math>\mathbf{r}</math>. According to the so-called "primitive model", these charges are distributed in a continuous medium that is characterized only by its [[relative static permittivity]], <math>\varepsilon_r</math>.
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| This distribution of charges within this medium gives rise to an [[electric potential]] <math>\Phi(\mathbf{r})</math> that satisfies [[Poisson's equation]]:
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| :<math> \nabla^2 \Phi(\mathbf{r}) = -\frac{1}{\varepsilon_r \varepsilon_0} \, \sum_{j = 1}^N q_j \, n_j(\mathbf{r})</math>,
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| where <math>\varepsilon_0</math> is the [[electric constant]].
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| The mobile charges not only establish <math>\Phi(\mathbf{r})</math> but also move in response to the associated [[Coulomb's law|Coulomb force]], <math>- q_j \, \nabla \Phi(\mathbf{r})</math>.
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| If we further assume the system to be in [[thermodynamic equilibrium]] with a [[heat bath]] at [[absolute temperature]] <math>T</math>, then the
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| concentrations of discrete charges, <math>n_j(\mathbf{r})</math>, may be considered to be
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| thermodynamic (ensemble) averages and the associated [[electric potential]] to be
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| a thermodynamic [[mean field theory|mean field]].
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| With these assumptions, the concentration of the <math>j</math>-th charge species is described
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| by the [[Boltzmann distribution]],
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| :<math> n_j(\mathbf{r}) = n_j^0 \, \exp\left( - \frac{q_j \, \Phi(\mathbf{r})}{k_B T} \right)</math>,
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| where <math>k_B</math> is [[Boltzmann's constant]] and where <math>n_j^0</math> is the mean
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| concentration of charges of species <math>j</math>.
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| Identifying the instantaneous concentrations
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| and potential in the Poisson equation with their mean-field counterparts in Boltzmann's distribution
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| yields the [[Poisson-Boltzmann equation]]:
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| :<math> \nabla^2 \Phi(\mathbf{r}) = -\frac{1}{\varepsilon_r \varepsilon_0} \, \sum_{j = 1}^N q_j n_j^0 \, \exp\left(- \frac{q_j \, \Phi(\mathbf{r})}{k_B T} \right)</math>.
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| Solutions to this nonlinear equation are known for some simple systems. Solutions for more general
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| systems may be obtained in the high-temperature (weak coupling) limit, <math>q_j \, \Phi(\mathbf{r}) \ll k_B T</math>, by [[Taylor expansion|Taylor expanding]] the exponential:
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| :<math> \exp\left(- \frac{q_j \, \Phi(\mathbf{r})}{k_B T} \right) \approx
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| 1 - \frac{q_j \, \Phi(\mathbf{r})}{k_B T}</math>.
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| This approximation yields the linearized Poisson-Boltzmann equation
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| :<math> \nabla^2 \Phi(\mathbf{r}) =
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| \left(\sum_{j = 1}^N \frac{n_j^0 \, q_j^2}{\varepsilon_r \varepsilon_0 \, k_B T} \right)\, \Phi(\mathbf{r}) - \frac{1}{\varepsilon_r \varepsilon_0} \, \sum_{j = 1}^N n_j^0 q_j
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| </math>
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| which also is known as the [[Debye-Hückel equation]]:<ref name=Kirby>{{cite book | author=Kirby BJ. | title=Micro- and Nanoscale Fluid Mechanics: Transport in Microfluidic Devices |
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| url=http://www.kirbyresearch.com/textbook}}</ref><ref name=DLi>{{cite book | author=Li D | title=Electrokinetics in Microfluidics |
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| year=2004}}</ref><ref name=Clemmow>{{cite book |title=Electrodynamics of particles and plasmas |url=http://books.google.com/books?id=SBNNzUrTjecC&pg=PP1&dq=particles+plasmas+inauthor:Clemmow#PPA236,M1|author=PC Clemmow & JP Dougherty|isbn=0-201-47986-9 |year=1969 |publisher=Addison-Wesley |location=Redwood City CA|pages=§7.6.7, p. 236 ff.}}</ref><ref name=Robinson>{{cite book |title=Electrolyte solutions |page=76 |url=http://books.google.com/books?id=6ZVkqm-J9GkC&pg=PR3#PPA76,M1 |author=RA Robinson &RH Stokes|isbn=0-486-42225-9|publisher=Dover Publications|location=Mineola NY |year=2002}}</ref><ref name=Brydges>See [http://www.springerlink.com/content/q2g78x6g72h21736/ DC Brydges & Ph A Martin '' Coulomb Systems at Low Density: A Review'']</ref>
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| The second term on the right-hand side vanishes for systems that are electrically neutral. The term in parentheses has the units of an inverse length squared and by
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| [[dimensional analysis]] leads to the definition of the characteristic length scale
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| :<math> \lambda_D =
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| \left(\frac{\varepsilon_r \varepsilon_0 \, k_B T}{\sum_{j = 1}^N n_j^0 \, q_j^2}\right)^{1/2}</math>
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| that commonly is referred to as the Debye-Hückel length. As the only characteristic length scale in the Debye-Hückel equation, <math>\lambda_D</math> 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.
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| The Debye-Hückel length may be expressed in terms of the [[Bjerrum length]] <math>\lambda_B</math> as
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| :<math> \lambda_D =
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| \left(4 \pi \, \lambda_B \, \sum_{j = 1}^N n_j^0 \, z_j^2\right)^{-1/2}</math>,
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| where <math>z_j = q_j/e</math> is the integer [[charge number]] that relates the charge on the <math>j</math>-th ionic
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| species to the [[elementary charge]] <math>e</math>.
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| ==Typical values==
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| In space plasmas where the electron density is relatively low, the Debye length may reach macroscopic values, such as in the magnetosphere, solar wind, interstellar medium and intergalactic medium (see table):
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| {| class="wikitable" style="margin: 1em auto"
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| |-
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| ! Plasma
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| ! Density<br>{{nobold|''n''<sub>e</sub>(m<sup>-3</sup>)}}
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| ! Electron temperature<br>{{nobold|''T''(K)}}
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| ! Magnetic field<br>{{nobold|''B''(T)}}
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| ! Debye length<br>{{nobold|''λ''<sub>D</sub>(m)}}
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| |- align=center
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| ! style="text-align:left" | Solar core
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| | 10<sup>32</sup>
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| | 10<sup>7</sup>
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| | --
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| | 10<sup>−11</sup>
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| |- align=center
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| ! style="text-align:left" | [[Tokamak]]
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| | 10<sup>20</sup>
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| | 10<sup>8</sup>
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| | 10
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| | 10<sup>−4</sup>
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| |- align=center
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| ! style="text-align:left" | Gas discharge
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| | 10<sup>16</sup>
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| | 10<sup>4</sup>
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| | --
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| | 10<sup>−4</sup>
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| |- align=center
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| ! style="text-align:left" | Ionosphere
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| | 10<sup>12</sup>
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| | 10<sup>3</sup>
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| | 10<sup>−5</sup>
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| | 10<sup>−3</sup>
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| |- align=center
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| ! style="text-align:left" | Magnetosphere
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| | 10<sup>7</sup>
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| | 10<sup>7</sup>
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| | 10<sup>−8</sup>
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| | 10<sup>2</sup>
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| |- align=center
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| ! style="text-align:left" | Solar wind
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| | 10<sup>6</sup>
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| | 10<sup>5</sup>
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| | 10<sup>−9</sup>
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| | 10
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| |- align=center
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| ! style="text-align:left" | Interstellar medium
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| | 10<sup>5</sup>
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| | 10<sup>4</sup>
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| | 10<sup>−10</sup>
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| | 10
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| |- align=center
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| ! style="text-align:left" | Intergalactic medium
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| | 1
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| | 10<sup>6</sup>
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| | --
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| | 10<sup>5</sup>
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| |-
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| | colspan=5 style="background:#fff; border-color:#aaa #fff #fff #fff; text-align:center" | Source: Chapter 19: ''The Particle Kinetics of Plasma''<br>
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| http://www.pma.caltech.edu/Courses/ph136/yr2004/
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| |}
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| [[Hannes Alfvén]] pointed out that: "In a low density plasma, localized space charge regions may build up large potential
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| drops over distances of the order of some tens of the Debye lengths. Such regions have been called ''electric double layers''. An electric [[Double layer (plasma)|double layer]] is the simplest space charge distribution that gives a potential drop in the layer and a vanishing electric field on each side of the layer. In the laboratory, double layers have been studied for half a century, but their importance in cosmic plasmas has not been generally recognized."{{fact|date=October 2013}}
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| == Debye length in a plasma ==
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| In a plasma, the background medium may be treated as the vacuum
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| (<math>\varepsilon_r = 1</math>), and the Debye length is
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| :<math> \lambda_D = \sqrt{\frac{\varepsilon_0 k_B/q_e^2}{n_e/T_e+\sum_{ij} j^2n_{ij}/T_i}}</math>
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| where
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| : λ<sub>''D''</sub> is the Debye length,
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| : ε<sub>0</sub> is the [[permittivity of free space]],
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| : ''k''<sub>''B''</sub> is the [[Boltzmann constant]],
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| : ''q''<sub>''e''</sub> is the charge of an [[electron]],
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| : ''T<sub>e</sub>'' and ''T<sub>i</sub>'' are the temperatures of the electrons and ions, respectively,
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| : ''n<sub>e</sub>'' is the density of electrons,
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| : ''n<sub>ij</sub>'' is the density of atomic species ''i'', with positive [[ion]]ic charge ''jq''<sub>''e''</sub>
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| The ion term is often dropped, giving
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| :<math> \lambda_D = \sqrt{\frac{\varepsilon_0 k_B T_e}{n_e q_e^2}}</math>
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| although this is only valid when the mobility of ions is negligible compared to the process's timescale.<ref>I. H. Hutchchinson - Principles of plasma diagnostics; ISBN 0-521-38583-0</ref>
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| == Debye length in an electrolyte ==
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| In an [[electrolyte]] or a [[colloids|colloidal suspension]], the Debye length<ref>Russel, W.B., Saville, D.A. and Schowalter, W.R. ''Colloidal Dispersions'', Cambridge University Press, 1989</ref> is usually denoted with symbol ''κ''<sup>−1</sup>
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| :<math> \kappa^{-1} = \sqrt{\frac{\varepsilon_r \varepsilon_0 k_B T}{2 N_A e^2 I}}</math>
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| where
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| : ''I'' is the [[ionic strength]] of the electrolyte, and here the unit should be mole/m<sup>3</sup>,
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| : ε<sub>0</sub> is the [[Vacuum permittivity|permittivity of free space]],
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| : ε<sub>r</sub> is the [[Relative static permittivity|dielectric constant]],
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| : ''k''<sub>B</sub> is the [[Boltzmann constant]],
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| : ''T'' is the absolute temperature in [[kelvin]]s,
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| : ''N<sub>A</sub>'' is the [[Avogadro number]].
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| : ''e'' is the [[elementary charge]],
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| or, for a symmetric monovalent electrolyte,
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| :<math> \kappa^{-1} = \sqrt{\frac{\varepsilon_r \varepsilon_0 R T}{2 F^2 C_0}}</math>
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| where
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| : ''R'' is the [[gas constant]],
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| : ''F'' is the [[Faraday constant]],
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| : ''C''<sub>0</sub> is the molar concentration of the electrolyte.
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| Alternatively,
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| :<math> \kappa^{-1} = \frac{1}{\sqrt{8\pi \lambda_B N_A I}} </math>
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| where
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| : <math>\lambda_B</math> is the [[Bjerrum length]] of the medium.
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| For water at room temperature, ''λ''<sub>B</sub> ≈ 0.7 nm.
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| At room temperature (25 °C), one can consider in water for 1:1 electrolytes the relation:<ref>Israelachvili, J., ''Intermolecular and Surface Forces'', Academic Press Inc., 1985, ISBN 0-12-375181-0</ref>
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| :<math> \kappa^{-1}(\mathrm{nm}) = \frac{0.304}{\sqrt{I(\mathrm{M})}}</math>
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| where
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| : ''κ''<sup>−1</sup> is expressed in [[nanometer]]s (nm)
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| : ''I'' is the [[ionic strength]] expressed in [[molar concentration|molar]] (M or mol/L)
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| == Debye length in semiconductors ==
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| The Debye length has become increasingly significant in the modeling of solid state devices as improvements in lithographic technologies have enabled smaller geometries.<ref>{{Cite journal| doi = 10.1021/nl071792z| volume = 7| issue = 11| pages = 3405–3409| last = Stern| first = Eric| coauthors = Robin Wagner, Fred J. Sigworth, Ronald Breaker, Tarek M. Fahmy, Mark A. Reed| title = Importance of the Debye Screening Length on Nanowire Field Effect Transistor Sensors| journal = Nano Letters| date = 2007-11-01|bibcode = 2007NanoL...7.3405S }}</ref><ref>
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| {{Cite journal| doi = 10.1063/1.118236| issn = 0003-6951| volume = 70| issue = 7| pages = 850| last = Guo| first = Lingjie| coauthors = Effendi Leobandung, Stephen Y. Chou| title = A room-temperature silicon single-electron metal–oxide–semiconductor memory with nanoscale floating-gate and ultranarrow channel| journal = Applied Physics Letters| accessdate = 2010-10-25| year = 1997| url = http://link.aip.org/link/APPLAB/v70/i7/p850/s1&Agg=doi|bibcode = 1997ApPhL..70..850G }}</ref><ref>
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| {{Cite journal| doi = 10.1063/1.117421| issn = 0003-6951| volume = 69| issue = 9| pages = 1232| last = Tiwari| first = Sandip| coauthors = Farhan Rana, Kevin Chan, Leathen Shi, Hussein Hanafi| title = Single charge and confinement effects in nano-crystal memories| journal = Applied Physics Letters| accessdate = 2010-10-25| year = 1996| url = http://link.aip.org/link/APPLAB/v69/i9/p1232/s1&Agg=doi|bibcode = 1996ApPhL..69.1232T }}</ref>
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|
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| The Debye length of [[semiconductors]] is given:
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| :<math> \mathit{L}_D = \sqrt{\frac{\varepsilon_{\mathrm{Si}} k_B T}{q^2N_d}}</math>
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| where
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| : ''ε''<sub>Si</sub> is the dielectric constant,
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| : ''k''<sub>B</sub> is the Boltzmann's constant,
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| : ''T'' is the absolute temperature in kelvins,
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| : ''q'' is the elementary charge, and
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| : ''N<sub>d</sub>'' is the density of dopants (either donors or acceptors).
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| When doping profiles exceed the Debye length, majority carriers no longer behave according to the distribution of the dopants. Instead, a measure of the profile of the doping gradients provides an “effective” profile that better matches the profile of the majority carrier density.
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| In the context of solids, the Debye length is also called the [[Thomas–Fermi screening|Thomas–Fermi screening length]].
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| ==See also==
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| [[Debye-Falkenhagen effect]]
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| ==References==
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| <references/>
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| ==Further reading==
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| * {{cite book | author=Goldston & Rutherford | title=Introduction to Plasma Physics | publisher=Institute of Physics Publishing, Philadelphia | year=1997 }}
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| * {{cite book | author=Lyklema | title=Fundamentals of Interface and Colloid Science | publisher=Academic Press, NY | year=1993 }}
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| {{DEFAULTSORT:Debye Length}}
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| [[Category:Electricity]]
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| [[Category:Colloidal chemistry]]
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| [[Category:Plasma physics]]
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| [[Category:Electrochemistry]]
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| [[Category:Length]]
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