Debye length

In plasma physics, the Debye length, named after the Dutch physical chemist Peter Debye, is the scale over which mobile charge carriers (e.g. electrons) screen out electric fields in plasmas and other conductors. In other words, the Debye length is the distance over which significant charge separation can occur. A Debye sphere is a volume whose radius is the Debye length, in which there is a sphere of influence, and outside of which charges are screened.

In space plasmas where the electron density is relatively low, the Debye length may reach macroscopic values, such as in the Magnetosphere, Solar wind, Interstellar medium and Intergalactic medium (see table):

Plasma	Density ne(m3)	Electron temperature T(K)	Magnetic field B(T)	Debye length λD(m)
Gas discharge	1016	104	—	10−4
Tokamak	1020	108	10	10−4
Ionosphere	1012	103	10−5	10−3
Magnetosphere	107	107	10−8	102
Solar core	1032	107	—	10−11
Solar wind	106	105	10−9	10
Interstellar medium	105	104	10−10	10
Intergalactic medium	1	106	—	105

Source: Chapter 19: The Particle Kinetics of Plasma
http://www.pma.caltech.edu/Courses/ph136/yr2002/Hannes Alfvén pointed out that: “In a low density plasma, localized space charge regions may build up large potential drops over distances of the order of some tens of the Debye lengths. Such regions have been called electric double layers. An electric double layer is the simplest space charge distribution that gives a potential drop in the layer and a vanishing electric field on each side of the layer. In the laboratory, double layers have been studied for half a century, but their importance in cosmic plasmas has not been generally recognized.”.

Debye length in a plasma

In a plasma, the Debye length is

\( \lambda_D = \sqrt{\frac{\varepsilon_0 k/q_e^2}{n_e/T_e+\sum_{ij} j^2n_{ij}/T_i}}\)

where

λ_D is the Debye length,

ε₀ is the permittivity of free space,

k is Boltzmann’s constant,

q_e is the charge on an electron,

T_e and T_i are the temperatures of the electrons and ions, respectively,

n_e is the density of electrons,

n_ij‘is the density of atomic species i, with positive ionic charge jq_e

The ion term is often dropped, giving

\( \lambda_D = \sqrt{\frac{\varepsilon_0 k T_e}{n_e q_e^2}}\)

although this is only valid when the ions are much colder than the electrons.

Debye length in an electrolyte

In an electrolyte, the Debye length is

\( \lambda_D = \sqrt{\frac{\varepsilon_0 \varepsilon_r k T}{2 N_A e^2 I}}\)

where

I is the ionic strength of the electrolyte,

ε₀ is the permittivity of free space,

ε_r is the dielectric constant,

k is the Boltzmann’s constant,

T is the Temperature,

N_A is Avogadro’s Number.

e is the elementary charge,

or when the solute is mono-monovalent and symmetrical,

\( \lambda_D = \sqrt{\frac{\varepsilon_0 \varepsilon_r R T}{2 F^2 C_0}}\)

where

R is the gas constant,

F is the Faraday constant,

C₀ is the molar concentration of the electrolyte.