Debye length

From Wikipedia, the free encyclopedia

The notion of Debye length plays an important role in plasma physics, electrolytes and colloids (DLVO theory).

In plasma physics, the Debye length (also called Debye radius), named after the Dutch physicist and 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.

1 Physical origin
2 Typical values
3 Debye length in a plasma
4 Debye length in an electrolyte
5 References
6 Further reading

[edit] Physical origin

The Debye length arises naturally in considering the screening of a source of electric potential by a cloud of charged particles whose density is determined by their energy in the electrical potential. If the potential to be screened is denoted by φ, the energy of a charged particle of charge q in this potential is qφ. It is convenient to take the charge q as the elementary charge of the electron. Assuming the likelihood of finding a particle with this energy is determined by a Boltzmann distribution, the number of particles at a location where the potential is φ becomes:

$N(\phi) = N_0 e^{-q \phi / (k_BT)} \ ,$

where N₀ is the density where the potential is zero, k_B is the Boltzmann constant and T is the absolute temperature in kelvins. It is assumed that the potential has the correct polarity to raise the energy of the charges screening the potential, which makes it less probable to find charges in regions of high potential. To determine the potential as a function of position, this charge density is placed in Poisson's equation to find:

${\nabla}^2 \phi (\mathbf{r} ) = \frac {qN_0\left(1- e^{-q \phi / (k_BT)}\right)}{\kappa \epsilon_0} \ ,$

where, as a mathematical convenience for purposes of illustration, the charge has been arranged to vanish when the potential is zero. The parameters in the Poisson equation are к = relative static electric permittivity of the medium, ε₀ = electric constant. A natural unit for potential is the thermal voltage defined as

$V_{th} = \frac {k_B T}{q} \ ,$

where q is the elementary charge. Using these units for potential, the Poisson equation becomes:

${\nabla}^2 \left(\frac {\phi (\mathbf{r} )}{V_{th}}\right) = \left(\frac {1}{\lambda_{D}}\right)^{2}\left(1-e^{-q \phi / (k_BT)}\right) \ ,$

with the Debye length λ_D defined by

$\lambda_D = \left(\frac {\kappa \epsilon_0 k_B T}{q^2 N_0}\right)^{1/2} \ .$

This Poisson equation is highly nonlinear. It can be solved in a one-dimensional case using an integrating factor, but to interpret the Debye length it suffices to take a simplified example with a very small potential (i.e., small compared to the thermal voltage). Then the equation can be linearized using a Taylor series for the exponential function to obtain the linear Debye-Hückel equation:^[1]^[2] ^[3]

${\nabla}^2 \left(\frac {\phi (\mathbf{r} )}{V_{th}}\right) = \left(\frac {1}{\lambda_{D}}\right)^{2}\left(\frac { \phi (\mathbf{r}) }{ V_{th}}\right) \ ,$

which has as solutions potentials decaying with distance from the originating potential at a rate given by the Debye length: the potential drops to 1/e of its unscreened value in approximately one Debye length. The rate of decay depends somewhat upon the dimensionality of the region of solution: for example, is the originating potential planar, cylindrical or spherical?

[edit] Typical values

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 n_e(m^-3)	Electron temperature T(K)	Magnetic field B(T)	Debye length λ_D(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 Alfven 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.".