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# 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:

$\Lambda = \frac {4\pi}{3} n \lambda_D^3$

where

n is the number density of particles,
λD is the Debye length.

Often the factor of 4π / 3 is dropped. When the Debye length is given by $\lambda_D = \sqrt{\frac{\epsilon_0 k T_e}{n_e q_e^2}}$, the plasma parameter is given by[1]:

$\Lambda = \frac{(\epsilon_0 k T_e)^{3/2}}{q_e^3 n_e^{1/2}}$

where

ε0 is the permittivity of free space,
k is Boltzmann's constant,
qe is the electron charge,
Teis the electron temperature.

The natural logarithm of the plasma parameter, lnΛ, is the Coulomb Logarithm.

## The plasma approximation

One of the criteria which determines whether a collection of charged particles can rigorously be termed a 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].

## 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 plasmasVery "cold" "high pressure" arc dischargeWhite dwarfs / neutron stars atmospheresPlasma ball Ionospheric physicsAstrophysical plasmasNuclear fusionSpace plasma physics

## 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