Debye length

In plasmas and electrolytes, the Debye length (also called Debye radius), named after Peter Debye, is a measure of a charge carrier's net electrostatic effect in a solution and how far its electrostatic effect persists. A Debye sphere is a volume whose radius is the Debye length. With each Debye length, charges are increasingly electrically screened. Every Debye‐length λ D {displaystyle lambda _{D}} , the electric potential will decrease in magnitude by 1/e. Debye length is an important parameter in plasma physics, electrolytes, and colloids (DLVO theory). The corresponding Debye screening wave vector k D = 1 / λ D {displaystyle k_{D}=1/lambda _{D}} for particles of density n {displaystyle n} , charge q {displaystyle q} at a temperature T {displaystyle T} is given by : k D 2 = 4 π n q 2 / k B T {displaystyle :k_{D}^{2}=4pi nq^{2}/k_{B}T} in Gaussian units. Expressions in MKS units will be given below. The analogous quantities at very low temperatures ( T → 0 {displaystyle T o 0} ) are known as the Thomas-Fermi length and the Thomas-Fermi wave vector. They are of interest in describing the behaviour of electrons in metals at room temperature.In a low density plasma, localized space charge regions may build up large potentialdrops 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. In plasmas and electrolytes, the Debye length (also called Debye radius), named after Peter Debye, is a measure of a charge carrier's net electrostatic effect in a solution and how far its electrostatic effect persists. A Debye sphere is a volume whose radius is the Debye length. With each Debye length, charges are increasingly electrically screened. Every Debye‐length λ D {displaystyle lambda _{D}} , the electric potential will decrease in magnitude by 1/e. Debye length is an important parameter in plasma physics, electrolytes, and colloids (DLVO theory). The corresponding Debye screening wave vector k D = 1 / λ D {displaystyle k_{D}=1/lambda _{D}} for particles of density n {displaystyle n} , charge q {displaystyle q} at a temperature T {displaystyle T} is given by : k D 2 = 4 π n q 2 / k B T {displaystyle :k_{D}^{2}=4pi nq^{2}/k_{B}T} in Gaussian units. Expressions in MKS units will be given below. The analogous quantities at very low temperatures ( T → 0 {displaystyle T o 0} ) are known as the Thomas-Fermi length and the Thomas-Fermi wave vector. They are of interest in describing the behaviour of electrons in metals at room temperature. The Debye length arises naturally in the thermodynamic description of large systems of mobile charges. In a system of N {displaystyle N} different species of charges, the j {displaystyle j} -th species carries charge q j {displaystyle q_{j}} and has concentration n j ( r ) {displaystyle n_{j}(mathbf {r} )} at position r {displaystyle mathbf {r} } . According to the so-called 'primitive model', these charges are distributed in a continuous medium that is characterized only by its relative static permittivity, ε r {displaystyle varepsilon _{r}} .This distribution of charges within this medium gives rise to an electric potential Φ ( r ) {displaystyle Phi (mathbf {r} )} that satisfies Poisson's equation: where ε ≡ ε r ε 0 {displaystyle varepsilon equiv varepsilon _{r}varepsilon _{0}} , ε 0 {displaystyle varepsilon _{0}} is the electric constant, and ρ E {displaystyle ho _{E}} is a charge density external (logically, not spatially) to the medium. The mobile charges not only establish Φ ( r ) {displaystyle Phi (mathbf {r} )} but also move in response to the associated Coulomb force, − q j ∇ Φ ( r ) {displaystyle -q_{j}, abla Phi (mathbf {r} )} .If we further assume the system to be in thermodynamic equilibrium with a heat bath at absolute temperature T {displaystyle T} , then the concentrations of discrete charges, n j ( r ) {displaystyle n_{j}(mathbf {r} )} , may be considered to be thermodynamic (ensemble) averages and the associated electric potential to be a thermodynamic mean field.With these assumptions, the concentration of the j {displaystyle j} -th charge species is describedby the Boltzmann distribution, where k B {displaystyle k_{B}} is Boltzmann's constant and where n j 0 {displaystyle n_{j}^{0}} is the meanconcentration of charges of species j {displaystyle j} . Identifying the instantaneous concentrations and potential in the Poisson equation with their mean-field counterparts in Boltzmann's distribution yields the Poisson–Boltzmann equation: Solutions to this nonlinear equation are known for some simple systems. Solutions for more generalsystems may be obtained in the high-temperature (weak coupling) limit, q j Φ ( r ) ≪ k B T {displaystyle q_{j},Phi (mathbf {r} )ll k_{B}T} , by Taylor expanding the exponential: