|
|
|
|
Debye length
|
| |
|
| |
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.
Discussion
Ask a question about 'Debye length' Start a new discussion about 'Debye length' Answer questions from other users |
Encyclopedia
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. The notion of Debye length plays an important role in plasma physics, electrolytes and colloids (DLVO theory).
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 f, the energy of a charged particle of charge q in this potential is qf. 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 f becomes:
where N0 is the density where the potential is zero, kB 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:
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 ? = relative static electric permittivity of the medium, e0 = electric constant. A natural unit for potential is the thermal voltage defined as
where q is the elementary charge. Using these units for potential, the Poisson equation becomes:
with the Debye length ?D defined by
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:
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 symmetry of the region in which the source of electric potential is localized: in the one-dimensional case (that is, for a planar charge distribution, as it occurs in p-n junctions usually analyzed in academic works), the characteristic length is exactly the Debye length.
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
ne(m-3) | Electron temperature
T(K) | Magnetic field
B(T) | Debye length
?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.".
Debye length in a plasma
In a plasma, the Debye length is
where
- ?D is the Debye length,
- e0 is the permittivity of free space,
- k is Boltzmann's constant,
- qe is the charge on an electron,
- Te and Ti are the temperatures of the electrons and ions, respectively,
- ne is the density of electrons,
- nijis the density of atomic species i, with positive ionic charge jqe
The ion term is often dropped, giving
although this is only valid when the ions are much colder than the electrons.
Debye length in an electrolyte
In an electrolyte or a colloidal dispersion, the Debye length is usually denoted with symbol ?-1
where
-
I is the ionic strength of the electrolyte, and here the unit should be mole/m3, e0 is the permittivity of free space, er is the dielectric constant, k is the Boltzmann's constant, T is the absolute temperature in kelvins, NA is Avogadro's number. e is the elementary charge,
or, for a symmetric monovalent electrolyte,
where
-
R is the gas constant, F is the Faraday constant, C0 is the molar concentration of the electrolyte.
Alternatively,
where
- is the Bjerrum length of the medium.
For water at room temperature, ?B ˜ 0.7 nm.
Further reading
|
| |
|
|
