Editing Debye length (section)

== In a plasma ==
For a weakly collisional plasma, Debye shielding can be introduced in a very intuitive way by taking into account the granular character of such a plasma. Let us imagine a sphere about one of its electrons, and compare the number of electrons crossing this sphere with and without Coulomb repulsion. With repulsion, this number is smaller. Therefore, according to Gauss theorem, the apparent charge of the first electron is smaller than in the absence of repulsion. The larger the sphere radius, the larger is the number of deflected electrons, and the smaller the apparent charge: this is Debye shielding. Since the global deflection of particles includes the contributions of many other ones, the density of the electrons does not change, at variance with the shielding at work next to a [[Langmuir probe]] ([[Debye sheath]]). Ions bring a similar contribution to shielding, because of the attractive Coulombian deflection of charges with opposite signs.

This intuitive picture leads to an effective calculation of Debye shielding (see section II.A.2 of <ref>{{ cite journal | last = Meyer-Vernet | first = N | year = 1993 | title = Aspects of Debye shielding | journal = American Journal of Physics | volume = 61 | pages = 249-257 }}</ref>). The assumption of a Boltzmann distribution is not necessary in this calculation: it works for whatever particle distribution function. The calculation also avoids approximating weakly collisional plasmas as continuous media. An N-body calculation reveals that the bare Coulomb acceleration of a particle by another one is modified by a contribution mediated by all other particles, a signature of Debye shielding (see section 8 of <ref>{{ cite journal | last1 = Escande | first1 = D. F. | last2 = Bénisti | first2 = D. | last3 = Elskens | first3 = Y. | last4 = Zarzoso | first4 = D. | last5 = Doveil | first5 = F. | year = 2018 | title = Basic microscopic plasma physics from N-body mechanics, A tribute to Pierre-Simon de Laplace | journal = Reviews of Modern Plasma Physics | volume = 2 | issue = 1 | page = 68 }}</ref>). When starting from random particle positions, the typical time-scale for shielding to set in is the time for a thermal particle to cross a Debye length, i.e. the inverse of the plasma frequency. Therefore in a weakly collisional plasma, collisions play an essential role by bringing a cooperative self-organization process: Debye shielding. This shielding is important to get a finite diffusion coefficient in the calculation of Coulomb scattering ([[Coulomb collision]]).

In a non-isothermic plasma, the temperatures for electrons and heavy species may differ while the background medium may be treated as the vacuum {{nowrap|(<math>\varepsilon_r = 1</math>),}} and the Debye length is
<math display="block"> \lambda_\text{D} = \sqrt{\frac{\varepsilon_0 k_\text{B}/q_e^2}{n_e/T_e+\sum_j z_j^2n_j/T_j}}</math>
where

* {{math|''λ''<sub>D</sub>}} is the Debye length,
* {{math|''ε''<sub>0</sub>}} is the [[permittivity of free space]],
* {{math|''k''<sub>B</sub>}} is the [[Boltzmann constant]],
* {{math|''q''<sub>''e''</sub>}} is the [[elementary charge|charge of an electron]],
* {{math|''T<sub>e</sub>''}} and {{math|''T<sub>i</sub>''}} are the temperatures of the electrons and ions, respectively,
* {{math|''n<sub>e</sub>''}} is the density of electrons,
* {{math|''n<sub>j</sub>''}} is the density of atomic species ''j'', with positive [[ion]]ic charge ''z<sub>j</sub>q<sub>e</sub>''

Even in quasineutral cold plasma, where ion contribution virtually seems to be larger due to lower ion temperature, the ion term is actually often dropped, giving
<math display="block"> \lambda_\text{D} = \sqrt{\frac{\varepsilon_0 k_\text{B} T_e}{n_e q_e^2}}</math>
although this is only valid when the mobility of ions is negligible compared to the process's timescale.<ref>{{cite book | first = I. H. | last = Hutchinson | title = Principles of plasma diagnostics | isbn = 0-521-38583-0 }}</ref>  A useful form of this equation is <ref name="chen">{{cite book|title=Introduction to Plasma Physics |year=1976|publisher=Plenum Press|last=Chen | first = Francis F.|page=10}}</ref>
<math display="block"> \lambda_\text{D} \approx 740 \sqrt{\frac{T_e}{n_e}}</math>
where <math>\lambda_\text{D}</math> is in cm, <math>T_e</math> in eV, and <math>n_e</math> in 1/cm<sup>3</sup>.

=== Typical values ===
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 the table here below:<ref>{{cite book | chapter=Chapter 20: The Particle Kinetics of Plasma |title=Applications of Classical Physics |author=Kip Thorne |date=2012 |url=http://www.pmaweb.caltech.edu/Courses/ph136/yr2012/ |chapter-url=http://www.pmaweb.caltech.edu/Courses/ph136/yr2012/1220.1.K.pdf |access-date=September 7, 2017}}</ref>

{| class="wikitable"
|-
! Plasma
! Density<br />{{nobold|''n''<sub>e</sub> (m<sup>−3</sup>)}}
! Electron temperature<br />{{nobold|''T'' (K)}}
! Magnetic field<br />{{nobold|''B'' (T)}}
! Debye length<br />{{nobold|''λ''<sub>D</sub> (m)}}
|- align=center
! style="text-align:left" | Solar core
| 10<sup>32</sup>
| 10<sup>7</sup>
| {{--}}
| 10<sup>−11</sup>
|- align=center
! style="text-align:left" | [[Tokamak]]
| 10<sup>20</sup>
| 10<sup>8</sup>
| 10
| 10<sup>−4</sup>
|- align=center
! style="text-align:left" | Gas discharge
| 10<sup>16</sup>
| 10<sup>4</sup>
| {{--}}
| 10<sup>−4</sup>
|- align=center
! style="text-align:left" | Ionosphere
| 10<sup>12</sup>
| 10<sup>3</sup>
| 10<sup>−5</sup>
| 10<sup>−3</sup>
|- align=center
! style="text-align:left" | Magnetosphere
| 10<sup>7</sup>
| 10<sup>7</sup>
| 10<sup>−8</sup>
| 10<sup>2</sup>
|- align=center
! style="text-align:left" | Solar wind
| 10<sup>6</sup>
| 10<sup>5</sup>
| 10<sup>−9</sup>
| 10
|- align=center
! style="text-align:left" | Interstellar medium
| 10<sup>5</sup>
| 10<sup>4</sup>
| 10<sup>−10</sup>
| 10
|- align=center
! style="text-align:left" | Intergalactic medium
| 1
| 10<sup>6</sup>
| {{--}}
| 10<sup>5</sup>
|-
|+ align="bottom" style="style="caption-side: bottom" | 
|}