Editing Debye length (section)

== Physical origin ==
[[File:Debye screening.svg|thumb|right|250px|Illustration of [[electric-field screening]] in a [[Plasma (physics)|plasma]]. The radius of the cloud of negative charge, rerpresented by the dashed circle, is the Debye length.]]

The Debye length arises naturally in the description of a substance with mobile charges, such as a [[Plasma (physics)|plasma]], [[Electrolyte|electrolyte solution]], or [[semiconductor]]. In such a substance, charges naturally [[Electric-field screening|screen out electric fields]] induced in the substance, with a certain [[characteristic length]].  That characteristic length is the Debye length.

Its value can be mathematically derived for a system of <math>N</math> different species of charged particles, where 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>.
The distribution of charged particles within this medium gives rise to an [[electric potential]] <math>\Phi(\mathbf{r})</math> that satisfies [[Poisson's equation]]:
<math display="block"> \varepsilon \nabla^2 \Phi(\mathbf{r}) = -\, \sum_{j = 1}^N q_j \, n_j(\mathbf{r}) - \rho_\text{ext}(\mathbf{r}),</math>
where <math>\varepsilon</math> is the medium's [[permitivity]], and <math>\rho_\text{ext}</math> is any static charge density that is not part of the medium.

The mobile charges don't only affect <math>\Phi(\mathbf{r})</math>, but are also affected by <math>\Phi(\mathbf{r})</math> due to the corresponding [[Coulomb's law|Coulomb force]], <math>- q_j \, \nabla \Phi(\mathbf{r})</math>.
If we further assume the system to be at temperature <math>T</math>, then the charge concentration <math>n_j(\mathbf{r})</math> may be considered, under the assumptions of [[mean field theory]], to tend toward the [[Boltzmann distribution]],
<math display="block"> n_j(\mathbf{r}) = n_j^0 \, \exp\left( - \frac{q_j \, \Phi(\mathbf{r})}{k_\text{B} T} \right),</math>
where <math>k_\text{B}</math> is the [[Boltzmann constant]] and where  <math>n_j^0</math> is the mean
concentration of charges of species <math>j</math>.

Identifying the instantaneous concentrations and potential in the Poisson equation with their mean-field counterparts in the Boltzmann distribution yields the [[Poisson–Boltzmann equation]]:
<math display="block"> \varepsilon \nabla^2 \Phi(\mathbf{r}) = -\, \sum_{j = 1}^N q_j n_j^0 \, \exp\left(- \frac{q_j \, \Phi(\mathbf{r})}{k_\text{B} T} \right) - \rho_\text{ext}(\mathbf{r}) .</math>

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, <math>q_j \, \Phi(\mathbf{r}) \ll k_\text{B} T</math>, by [[Taylor expansion|Taylor expanding]] the exponential:
<math display="block"> \exp\left(- \frac{q_j \, \Phi(\mathbf{r})}{k_\text{B} T} \right) \approx
1 - \frac{q_j \, \Phi(\mathbf{r})}{k_\text{B} T}.</math>

This approximation yields the linearized Poisson–Boltzmann equation
<math display="block"> \varepsilon \nabla^2 \Phi(\mathbf{r}) =
\left(\sum_{j = 1}^N \frac{n_j^0 \, q_j^2}{ k_\text{B} T} \right)\, \Phi(\mathbf{r}) -\, \sum_{j = 1}^N n_j^0 q_j - \rho_\text{ext}(\mathbf{r})
</math>
which also is known as the [[Debye–Hückel equation]]:<ref name=Kirby>{{cite book |last=Kirby |first=B. J. |title=Micro- and Nanoscale Fluid Mechanics: Transport in Microfluidic Devices |location=New York |publisher=Cambridge University Press |year=2010 |isbn=978-0-521-11903-0 }}</ref><ref name=DLi>{{cite book |last=Li |first=D. | title=Electrokinetics in Microfluidics |publisher=Academic Press |isbn=0-12-088444-5 |year=2004 }}</ref><ref name=Clemmow>{{cite book |title=Electrodynamics of particles and plasmas |url=https://books.google.com/books?id=SBNNzUrTjecC&q=particles+plasmas+inauthor:Clemmow&pg=PP1 |author=PC Clemmow & JP Dougherty |isbn=978-0-201-47986-7 |year=1969 |publisher=[[Addison-Wesley]] |location=Redwood City CA |pages=§&nbsp;7.6.7, p. 236 ff }}{{Dead link|date=January 2024 |bot=InternetArchiveBot |fix-attempted=yes }}</ref><ref name=Robinson>{{cite book |title=Electrolyte solutions |page=76 |url=https://books.google.com/books?id=6ZVkqm-J9GkC&pg=PR3 |author=RA Robinson &RH Stokes| isbn=978-0-486-42225-1 |publisher=[[Dover Publications]] |location=Mineola, NY |year=2002}}</ref><ref name=Brydges>See {{cite journal| last1=Brydges|first1=David C.| last2=Martin|first2=Ph. A.|journal=Journal of Statistical Physics|volume=96|issue=5/6| year=1999|pages=1163–1330|doi=10.1023/A:1004600603161|title=Coulomb Systems at Low Density: A Review|arxiv = cond-mat/9904122 |bibcode = 1999JSP....96.1163B |s2cid=54979869}}</ref>
The second term on the right-hand side vanishes for systems that are electrically neutral. The term in parentheses divided by <math>\varepsilon</math> has the units of an inverse length squared, and by
[[dimensional analysis]] leads to the definition of the characteristic length scale:
{{Equation box 1
|title='''Debye length'''
|indent=:
|equation=<math> \lambda_\text{D} =
\left(\frac{\varepsilon \, k_\text{B} T}{\sum_{j = 1}^N n_j^0 \, q_j^2}\right)^{1/2}</math>
}}
Substituting this length scale into the Debye–Hückel equation and neglecting the second and third terms on the right side yields the much simplified form <math> \lambda_\text{D}^2 \nabla^2 \Phi(\mathbf{r}) = \Phi(\mathbf{r}) </math>. As the only characteristic length scale in the Debye–Hückel equation, <math>\lambda_\text{D}</math> sets the scale for variations in the potential and in the concentrations of charged species. All charged species contribute to the Debye length in the same way, regardless of the sign of their charges.

To illustrate Debye screening, one can consider the example of a point charge placed in a plasma. The external charge density is then <math>\rho_\text{ext} = Q\delta(\mathbf{r})</math>, and the resulting potential is
<math display="block"> \Phi(\mathbf{r}) = \frac{Q}{4\pi\varepsilon r} e^{-r/\lambda_\text{D}}</math>
The bare Coulomb potential is exponentially screened by the medium, over a distance of the Debye length: this is called [[Electric-field screening|Debye screening]] or shielding.

The Debye length may be expressed in terms of the [[Bjerrum length]] <math>\lambda_\text{B}</math> as
<math display="block"> \lambda_\text{D} =
\left(4 \pi \, \lambda_\text{B} \, \sum_{j = 1}^N n_j^0 \, z_j^2\right)^{-1/2},</math>
where <math>z_j = q_j/e</math> is the integer [[charge number]] that relates the charge on the <math>j</math>-th ionic
species to the [[elementary charge]] <math>e</math>.