Plasma parameter: Difference between revisions

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{{Distinguish|Plasma parameters}}

The '''plasma parameter''' is a [[dimensionless quantity|dimensionless number]], denoted by capital Lambda, {{math|Λ}}. The plasma parameter is usually interpreted to be the argument of the Coulomb logarithm, which is the ratio of the maximum impact parameter to the classical distance of closest approach in [[Coulomb collision|Coulomb scattering]]. In this case, the plasma parameter is given by:<ref>{{ cite book | last = Chen, | first = Francis F.F., | title = Introduction to Plasma Physics and Controlled Fusion, (| publisher = Springer, | ___location = New York, | year = 2006) }}</ref>

:<math> display="block">\Lambda = 4\pi n n_\lambda_Dtext{e}\lambda_\text{D}^3 </math>

:<math> \Lambda = 4\pi n \lambda_D^3 </math>

where

:* {{math|''n''_e}} is the [[number density]] of electrons,

:* {{math|''λ''_D}} is the [[Debye length]].

This expression is typically valid for a plasma in which ion thermal velocities are much less than electron thermal velocities. A detailed discussion of the Coulomb logarithm is available in the ''NRL Plasma Formulary'', pages 34–35.

Note that the word parameter is usually used in plasma physics to refer to bulk plasma properties in general: see [[plasma parameters]].

An alternative definition of this parameter is given by the average number of electrons in a [[plasma (physics)|plasma]] contained within a [[Debye sphere]] (a sphere of radius the [[Debye length]]). This definition of the plasma parameter is more frequently (and appropriately) called the Debye number, and is denoted <math>N_\text{D}</math>. In this context, the plasma parameter is defined as

<math display="block">N_\text{D} = \frac{4\pi}{3} n_\text{e}\lambda_\text{D}^3 = \frac{1}{3}\Lambda</math>

This definition of the plasma parameter is more frequently (and appropriately) called the Debye number, and is denoted <math> N_D</math>. In this context, the plasma parameter is defined as

:<math> N_D = \frac {4\pi}{3} n \lambda_D^3 </math>

Since these two definitions differ only by a factor of three, they are frequently used interchangeably.

Often the factor of <math>\frac{4\pi/}{3}</math> is dropped. When the Debye length is given by <math> \lambda_Dlambda_\text{D} = \sqrt{\frac{\epsilon_0varepsilon_0 k T_ek_\text{B}T_\text{n_e q_ee}}{n_\text{e}q_\text{e}^2}}</math>, the plasma parameter is given by<ref>{{cite book | last = Miyamoto, | first = K., | title = Fundamentals of Plasma Physics and Controlled Fusion, (| ___location = Iwanami, Tokyo, | year = 1997)}}</ref>

<math display="block">N_\text{D} = \frac{{\left(\varepsilon_0 k_\text{B} T_\text{e}\right)}^{3/2}}{q_\text{e}^3 {n_\text{e}}^{1/2}}

:<math> N_D = \left(\frac{(k_\epsilon_0 ktext{B} T_e}{n_e^{1/3}}\right)^{3/2}} \left(\frac{q_e^3 n_e2}{\varepsilon_0}\right)^{1-3/2}} </math>

where

:* {{math|''ε''₀}} is the [[permittivity of free space]],

:* {{math|''k''_B}} is the [[Boltzmann's constant]],

:* {{math|''q''_''e''}} is the electron charge,

:* {{math|''T''_e''}} is the electron temperature.

Confusingly, some authors define the plasma parameter as :

:<math> display="block">\epsilon_pvarepsilon_p = \Lambda^{-1}\ .</math>.

:<math> \epsilon_p = \Lambda^{-1}\ </math>.

== Coupling parameter ==

A closely related parameter is the plasma coupling <math>\Gamma</math>, defined as a ratio of the Coulomb energy to the thermal one:

:<math> display="block">\Gamma = \frac{E_\mathrmtext{C}}{kT_ek_\text{B} T_\text{e}}.</math>.

:<math> \Gamma = \frac{E_\mathrm{C}}{kT_e} </math>.

The Coulomb energy (per particle) is

:<math> display="block">E_\mathrmtext{C} = \frac{q_eq_\text{e}^2}{4\pi\epsilon_0varepsilon_0\langle r \rangle},</math>,

where for the typical inter-particle distance <math>\langle r \rangle</math> usually is taken the [[Wigner-SeitzWigner–Seitz radius]]. Therefore,

:<math> E_\mathrm{C} = \frac{q_e^2}{4\pi\epsilon_0\langle r \rangle}</math>,

:<math> display="block">\Gamma = \frac{q_eq_\text{e}^2}{4\pi\epsilon_0varepsilon_0 kT_ek_\text{B}T_\text{e}}\sqrt[3]{\frac{4\pi n_en_\text{e}}{3}} .</math>.

where for the typical inter-particle distance <math>\langle r \rangle</math> usually is taken the [[Wigner-Seitz radius]]. Therefore,

:<math> \Gamma = \frac{q_e^2}{4\pi\epsilon_0 kT_e}\sqrt[3]{\frac{4\pi n_e}{3}} </math>.

Clearly, up to a numeric factor of the order of unity,

:<math> display="block">\Gamma \sim \Lambda^{-2/3}\ .</math>.

:<math> \Gamma \sim \Lambda^{-2/3}\ </math>.

In general, for multicomponent plasmas one defines the coupling parameter for each species ''s'' separately:

:<math> display="block">\Gamma_s = \frac{q_s^2}{4\pi \epsilon_0varepsilon_0 kT_sk_\text{B}T_s} \sqrt[3]{\frac{4\pi n_s}{3}} .</math>.

:<math> \Gamma_s = \frac{q_s^2}{4\pi\epsilon_0 kT_s}\sqrt[3]{\frac{4\pi n_s}{3}} </math>.

Here, ''s'' stands for either electrons or (a type of) ions.

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== The ideal plasma approximation ==

One of the criteria which determine whether a collection of charged particles can rigorously be termed an [[ideal plasma]] is that {{math|Λ ≫ 1}}. When this is the case, collective electrostatic interactions dominate over binary collisions, and the plasma particles can be treated as if they only interact with a smooth background field, rather than through pairwise interactions (collisions).<ref>J.D. Callen, University of Wisconsin-Madison, Draft Material for Fundamentals of Plasma Physics book: Collective Plasma Phenomena [http://homepages.cae.wisc.edu/~callen/chap1.pdf PDF]</ref> The [[equation of state]] of each species in an ideal plasma is that of an [[ideal gas]].

One of the criteria which determine whether a collection of charged particles can rigorously be termed an [[ideal plasma]] is that Λ>>1.

When this is the case, collective electrostatic interactions dominate over binary collisions, and the plasma particles can be treated as if they only interact with a smooth background field, rather than through pairwise interactions (collisions).<ref>J.D. Callen, University of Wisconsin-Madison, Draft Material for Fundamentals of Plasma Physics book: Collective Plasma Phenomena [http://homepages.cae.wisc.edu/~callen/chap1.pdf PDF]</ref> The [[equation of state]] of each species in an ideal plasma is that of an [[ideal gas]].

== Plasma properties and Λ ==

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{| class="wikitable" style="text-align:center"

|-

! rowspan=2 valign=bottom | Description

! colspan=2 | Plasma parameter magnitude

|-

! <{{math>\Lambda|Λ \ll≪ 1 ~}} (\Gamma{{math|Γ \gg≫ 1}})</math> || <{{math>\Lambda|Λ \gg≫ 1 ~}} (\Gamma{{math|Γ \ll≪ 1}})</math>

|-

! Coupling

| Strongly coupled plasma || Weakly coupled plasma

|-

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! Typical characteristic

| Cold and dense || Hot and diffuse

|- style="vertical-align:top;"

|-

! Examples

| Solid-density laser ablation plasmas<br>Very "cold" "high pressure" arc discharge<br>Inertial fusion experiments<br>WhiteStellar dwarfs / neutron stars atmospheres |interiors
| Ionospheric physics<br>Magnetic fusion devices<br>Space plasma physics<br>Plasma ball

|}

== References ==

{{reflist}}

== External links ==

* [https://web.archive.org/web/20090325194948/http://wwwppd.nrl.navy.mil/nrlformulary/NRL_FORMULARY_07.pdf NRL Plasma Formulary 2007 ed.]

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[[Category:Plasma physics| parameters]]