Plasma parameter

The plasma parameter is a 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 scattering. In this case, the plasma parameter is given by:^[1] $${\displaystyle \Lambda =4\pi n_{\text{e}}\lambda _{\text{D}}^{3}}$$ where

• n_e 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.

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 ${\displaystyle N_{\text{D}}}$. In this context, the plasma parameter is defined as $${\displaystyle N_{\text{D}}={\frac {4\pi }{3}}n_{\text{e}}\lambda _{\text{D}}^{3}={\frac {1}{3}}\Lambda }$$

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

Often the factor of ${\displaystyle {\frac {4\pi }{3}}}$ is dropped. When the Debye length is given by ${\displaystyle \lambda _{\text{D}}={\sqrt {\frac {\epsilon _{0}kT_{\text{e}}}{n_{\text{e}}q_{\text{e}}^{2}}}}}$, the plasma parameter is given by^[2] $${\displaystyle N_{\text{D}}={\frac {(\epsilon _{0}kT_{\text{e}})^{\frac {3}{2}}}{q_{\text{e}}^{3}{n_{\text{e}}}^{\frac {1}{2}}}}}$$ 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: $${\displaystyle \epsilon _{p}=\Lambda ^{-1}\ .}$$