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:

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

In it is

• λ _{D is} the Debye length
• λ _{De is} the electron Debye length
• λ _{Di is} the ion debye length (for singly charged ions)
• T _e , T _{i is} the temperature of the electrons or ions
• n _e , the particle density of the electron
• ε _{0 is} the electric field constant
• k _{B is} the Boltzmann constant
• e is the elementary charge, i.e. the charge of an electron.

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. ${\displaystyle p_{0}\,}$