Plasma parameter

The plasma parameter is a dimensionless number, denoted by capital Lambda, Λ, which measures the average number of electrons contained within a Debye sphere (a sphere of radius the Debye length) in a plasma (but note that the word parameter is usually used in plasma physics to refer to bulk plasma properties in general: see plasma parameters). It is defined as:

${\displaystyle \Lambda ={\frac {4\pi }{3}}n\lambda _{D}^{3}}$

where

n is the number density of particles,

λ_D is the Debye length.

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

${\displaystyle \Lambda ={\frac {(\epsilon _{0}kT_{e})^{3/2}}{q_{e}^{3}n_{e}^{1/2}}}}$

where

ε₀ is the permittivity of free space,

k is Boltzmann's constant,

q_e is the electron charge,

T_eis the electron temperature.

Confusingly, some authors define the plasma parameter as :

${\displaystyle \epsilon _{p}=\Lambda ^{-1}\ }$.

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 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) ^[2]. The equation of state of ideal plasma is that of ideal gas.

Plasma properties and Λ

The magnitude of Λ can be summarised below ^[3]:

Description	Plasma parameter magnitude
	Λ<<1	Λ>>1
Coupling	Strongly coupled plasma	Weakly coupled plasma
Debye sphere	Sparsely populated	Densely populated
Electrostatic influence	Almost continuously	Occasional
Typical characteristic	Cold and dense	Hot and diffuse
Examples	Solid-density laser ablation plasmas Very "cold" "high pressure" arc discharge Inertial fusion experiments White dwarfs / neutron stars atmospheres	Ionospheric physics Magnetic fusion devices Space plasma physics Plasma ball

References

1. Miyamoto, K., Fundamentals of Plasma Physics and Controlled Fusion, (Iwanami, Tokyo, 1997)
2. J.D. Callen, University of Wisconsin-Madison, Draft Material for Fundamentals of Plasma Physics book: Collective Plasma Phenomena PDF
3. See The plasma parameter lecture notes from Richard Fitzpatrick