Plasma parameter: Difference between revisions

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The '''plasma parameter''' is a [[dimensionless quantity|dimensionless number]], denoted by capital Lambda, 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>Chen, F.F., Introduction to Plasma Physics and Controlled Fusion, (Springer, New York, 2006)</ref>

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

:<math>\Lambda = 4\pi n\lambda_\text{D}^3</math>

where

:* ''n'' is the [[number density]] of electrons,

:* 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.

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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\lambda_\text{D}^3 = \frac{1}{3}\Lambda</math>

:<math>N_\text{D} = \frac{4\pi}{3} n\lambda_\text{D}^3 = \frac{1}{3}\Lambda</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_\text{D} = \sqrt{\frac{\epsilon_0 kT_\text{e}}{n_\text{e}q_\text{e}^2}}</math>, the plasma parameter is given by<ref>Miyamoto, K., Fundamentals of Plasma Physics and Controlled Fusion, (Iwanami, Tokyo, 1997)</ref>

:<math display="block">N_\text{D} = \frac{(\epsilon_0 kT_\text{e})^\frac{3}{2}}{q_\text{e}^3 {n_\text{e}}^\frac{1}{2}}</math>

:<math>N_\text{D} = \frac{(\epsilon_0 kT_\text{e})^\frac{3}{2}}{q_\text{e}^3 {n_\text{e}}^\frac{1}{2}}</math>

where

:* ε₀ is the [[permittivity of free space]],

:* ''k'' is the [[Boltzmann constant]],

:* ''q''_e is the electron charge,

:* ''T''_e is the electron temperature.

Confusingly, some authors define the plasma parameter as :

:<math display="block">\epsilon_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_\text{C}}{kT_\text{e}}.</math>.

:<math>\Gamma = \frac{E_\text{C}}{kT_\text{e}}</math>.

The Coulomb energy (per particle) is

:<math display="block">E_\text{C} = \frac{q_\text{e}^2}{4\pi\epsilon_0\langle r \rangle},</math>,

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

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

:<math display="block">\Gamma = \frac{q_\text{e}^2}{4\pi\epsilon_0 kT_\text{e}}\sqrt[3]{\frac{4\pi n_\text{e}}{3}}.</math>.

:<math>\Gamma = \frac{q_\text{e}^2}{4\pi\epsilon_0 kT_\text{e}}\sqrt[3]{\frac{4\pi n_\text{e}}{3}}</math>.

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

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

:<math>\Gamma \sim \Lambda^{-\frac{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_0 kT_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.