plasma parameter

{{Distinguish|Plasma parameters}}

The plasma parameter is a 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 scattering. In this case, the plasma parameter is given by:{{ cite book | last = Chen | first = Francis F. | title = Introduction to Plasma Physics and Controlled Fusion | publisher = Springer | location = New York | year = 2006 }}

$$\backslash Lambda\; =\; 4\backslash pi\; n\_\backslash text\{e\}\backslash lambda\_\backslash text\{D\}^3$$

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 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 $N\_\backslash text\{D\}$. In this context, the plasma parameter is defined as

$$N\_\backslash text\{D\}\; =\; \backslash frac\{4\backslash pi\}\{3\}\; n\_\backslash text\{e\}\backslash lambda\_\backslash text\{D\}^3\; =\; \backslash frac\{1\}\{3\}\backslash Lambda$$

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

Often the factor of $\backslash frac\{4\backslash pi\}\{3\}$ is dropped. When the Debye length is given by $\backslash lambda\_\backslash text\{D\}\; =\; \backslash sqrt\{\backslash frac\{\backslash varepsilon\_0\; k\_\backslash text\{B\}T\_\backslash text\{e\}\}\{n\_\backslash text\{e\}q\_\backslash text\{e\}^2\}\}$, the plasma parameter is given by{{cite book | last = Miyamoto | first = K. | title = Fundamentals of Plasma Physics and Controlled Fusion | location = Iwanami, Tokyo | year = 1997}}

$$N\_\backslash text\{D\}\; =\; \backslash frac\{\{\backslash left(\backslash varepsilon\_0\; k\_\backslash text\{B\}\; T\_\backslash text\{e\}\backslash right)\}^\{3/2\}\}\{q\_\backslash text\{e\}^3\; \{n\_\backslash text\{e\}\}^\{1/2\}\}$$

= \left(\frac{k_\text{B} T_e}{n_e^{1/3}}\right)^{3/2} \left(\frac{q_e^2}{\varepsilon_0}\right)^{-3/2}

where

• {{math|ε₀}} is the permittivity of free space,
• {{math|k_B}} is the Boltzmann constant,
• {{math|q_e}} is the electron charge,
• {{math|T_e}} is the electron temperature.

Confusingly, some authors define the plasma parameter as:

$$\backslash varepsilon\_p\; =\; \backslash Lambda^\{-1\}\; .$$