# Debye length

In the plasma physics is screening length by Peter Debye Debye length or Debye radius called the characteristic length on which the electric potential of a local excess charge on the fold decreases ( : Euler's number ). ${\ displaystyle \ lambda _ {\ mathrm {D}},}$ ${\ displaystyle {\ tfrac {1} {e}}}$${\ displaystyle e}$

Ion distribution in a solution

Due to the electrostatic repulsion or attraction, there are, on a statistical average, fewer charge carriers of the same polarity than those of opposite polarity in the immediate vicinity of a charge . This shields the load from the outside (see illustration). The thermal movement of the particles disrupts the order and thus weakens the shielding effect. The resulting shielding length is a central variable in the Debye-Hückel theory . Under given conditions, its value depends on the symmetry of the problem: one speaks of shielding length in the case of a plane charge distribution, of Debye-Radius with spherical symmetry .

The principle of shielding a charge by freely moving charge carriers is applicable to plasmas , electrolytes and semiconductors .

## Debye length in plasmas

The following applies in equilibrium:

{\ displaystyle {\ begin {aligned} {\ lambda _ {\ mathrm {D}}} ^ {- 2} & = {\ lambda _ {\ mathrm {De}}} ^ {- 2} + {\ lambda _ {\ mathrm {Tue}}} ^ {- 2} \\ & = {\ frac {n_ {e} \, e ^ {2}} {\ varepsilon _ {0}}} \ left ({\ frac {1 } {k _ {\ mathrm {B}} \, T_ {e}}} + {\ frac {1} {k _ {\ mathrm {B}} \, T_ {i}}} \ right) \ end {aligned} }}

In it is

In a plasma with a low particle density, the electrons are often much hotter than the ions in the presence of electric fields and are therefore more evenly distributed. Then:

{\ displaystyle {\ begin {aligned} T_ {e} & \ gg T_ {i} \\\ Leftrightarrow {\ frac {1} {k _ {\ mathrm {B}} \, T_ {e}}} & \ ll {\ frac {1} {k _ {\ mathrm {B}} \, T_ {i}}} \\\ Rightarrow \ lambda _ {\ mathrm {D}} & \ approx \ lambda _ {\ mathrm {Di}} = {\ sqrt {\ frac {\ varepsilon _ {0} \, k _ {\ mathrm {B}} T_ {i}} {n_ {e} \, e ^ {2}}}} \ end {aligned}} }

Conversely, in a dense plasma or with rapidly changing fields, the mobility of the ions is too low to adapt their density to the field. Then the ion term can be neglected:

${\ displaystyle \ lambda _ {\ mathrm {D}} \ approx \ lambda _ {\ mathrm {De}} = {\ sqrt {\ frac {\ varepsilon _ {0} \, k _ {\ mathrm {B}} T_ {e}} {n_ {e} \, e ^ {2}}}}}$.

## Debye length in electrolytes

The following applies in electrolytes:

${\ displaystyle \ lambda _ {\ mathrm {D}} = {\ sqrt {\ frac {\ varepsilon _ {0} \, \ varepsilon _ {\ mathrm {r}} \, k _ {\ mathrm {B}} \ , T} {2 \, N _ {\ mathrm {A}} \, e ^ {2} \, I}}}}$,

in which

• ${\ displaystyle \ varepsilon _ {\ mathrm {r}}}$the permittivity
• ${\ displaystyle N _ {\ mathrm {A}}}$the Avogadro constant
• I is the ionic strength of the electrolyte.

For aqueous solutions of a 1: 1 electrolyte such as common salt , the Debye length in 0.1 molar solution is 0.96  nm at room temperature , and 9.6 nm in 0.001 molar solution.

## Debye length in semiconductors

For an n-type semiconductor :

${\ displaystyle \ lambda _ {\ mathrm {Dn}} = {\ sqrt {\ frac {\ varepsilon \, U_ {T}} {e \, n_ {0}}}}}$

and for a p-type semiconductor:

${\ displaystyle \ lambda _ {\ mathrm {Dp}} = {\ sqrt {\ frac {\ varepsilon \, U_ {T}} {e \, p_ {0}}}}}$

It is

• ${\ displaystyle \ varepsilon = \ varepsilon _ {0} \, \ varepsilon _ {\ mathrm {r}}}$the dielectric constant of the semiconductor
• ${\ displaystyle U_ {T} = {\ frac {k _ {\ mathrm {B}} \, T} {e}}}$the temperature stress
• ${\ displaystyle n_ {0} \,}$or the equilibrium charge carrier density of the semiconductor.${\ displaystyle p_ {0} \,}$