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In plasmas and electrolytes, the Debye length \lambda_ (also called Debye radius), is a measure of a
charge carrier In physics, a charge carrier is a particle or quasiparticle that is free to move, carrying an electric charge, especially the particles that carry electric charges in electrical conductors. Examples are electrons, ions and holes. The term is used ...
's net electrostatic effect in a
solution Solution may refer to: * Solution (chemistry), a mixture where one substance is dissolved in another * Solution (equation), in mathematics ** Numerical solution, in numerical analysis, approximate solutions within specified error bounds * Solutio ...
and how far its electrostatic effect persists. With each Debye length the charges are increasingly electrically screened and the electric potential decreases in magnitude by 1/ e. A Debye sphere is a volume whose radius is the Debye length. Debye length is an important parameter in plasma physics,
electrolytes An electrolyte is a medium containing ions that is electrically conducting through the movement of those ions, but not conducting electrons. This includes most soluble salts, acids, and bases dissolved in a polar solvent, such as water. Upon dis ...
, and
colloids A colloid is a mixture in which one substance consisting of microscopically dispersed insoluble particles is suspended throughout another substance. Some definitions specify that the particles must be dispersed in a liquid, while others extend ...
(
DLVO theory The DLVO theory (named after Boris Derjaguin and Lev Landau, Evert Verwey and Theodoor Overbeek) explains the aggregation of aqueous dispersions quantitatively and describes the force between charged surfaces interacting through a liquid medium ...
). The corresponding Debye screening wave vector k_=1/\lambda_ for particles of density n, charge q at a temperature T is given by k_^2=4\pi n q^2/(k_T) in
Gaussian units Gaussian units constitute a metric system of physical units. This system is the most common of the several electromagnetic unit systems based on cgs (centimetre–gram–second) units. It is also called the Gaussian unit system, Gaussian-cgs unit ...
. Expressions in MKS units will be given below. The analogous quantities at very low temperatures (T \to 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 is named after the Dutch-American physicist and chemist Peter Debye (1884-1966), a Nobel laureate in Chemistry.


Physical origin

The Debye length arises naturally in the thermodynamic description of large systems of mobile charges. In a system of N different species of charges, the j-th species carries charge q_j and has concentration n_j(\mathbf) at position \mathbf. According to the so-called "primitive model", these charges are distributed in a continuous medium that is characterized only by its
relative static permittivity The relative permittivity (in older texts, dielectric constant) is the permittivity of a material expressed as a ratio with the electric permittivity of a vacuum. A dielectric is an insulating material, and the dielectric constant of an insul ...
, \varepsilon_r. This distribution of charges within this medium gives rise to an electric potential \Phi(\mathbf) that satisfies
Poisson's equation Poisson's equation is an elliptic partial differential equation of broad utility in theoretical physics. For example, the solution to Poisson's equation is the potential field caused by a given electric charge or mass density distribution; with th ...
: \varepsilon \nabla^2 \Phi(\mathbf) = -\, \sum_^N q_j \, n_j(\mathbf) - \rho_(\mathbf), where \varepsilon \equiv \varepsilon_r \varepsilon_0, \varepsilon_0 is the electric constant, and \rho_ is a charge density external (logically, not spatially) to the medium. The mobile charges not only contribute in establishing \Phi(\mathbf) but also move in response to the associated
Coulomb force Coulomb's inverse-square law, or simply Coulomb's law, is an experimental law of physics that quantifies the amount of force between two stationary, electrically charged particles. The electric force between charged bodies at rest is convention ...
, - q_j \, \nabla \Phi(\mathbf). If we further assume the system to be in
thermodynamic equilibrium Thermodynamic equilibrium is an axiomatic concept of thermodynamics. It is an internal state of a single thermodynamic system, or a relation between several thermodynamic systems connected by more or less permeable or impermeable walls. In ther ...
with a heat bath at
absolute temperature Thermodynamic temperature is a quantity defined in thermodynamics as distinct from kinetic theory or statistical mechanics. Historically, thermodynamic temperature was defined by Kelvin in terms of a macroscopic relation between thermodynamic wo ...
T, then the concentrations of discrete charges, n_j(\mathbf), 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-th charge species is described by the
Boltzmann distribution In statistical mechanics and mathematics, a Boltzmann distribution (also called Gibbs distribution Translated by J.B. Sykes and M.J. Kearsley. See section 28) is a probability distribution or probability measure that gives the probability t ...
, n_j(\mathbf) = n_j^0 \, \exp\left( - \frac \right), where k_ is the
Boltzmann constant The Boltzmann constant ( or ) is the proportionality factor that relates the average relative kinetic energy of particles in a gas with the thermodynamic temperature of the gas. It occurs in the definitions of the kelvin and the gas constan ...
and where n_j^0 is the mean concentration of charges of species j. Identifying the instantaneous concentrations and potential in the Poisson equation with their mean-field counterparts in the Boltzmann distribution yields the
Poisson–Boltzmann equation The Poisson–Boltzmann equation is a useful equation in many settings, whether it be to understand physiology, physiological interfaces, polymer science, electron interactions in a semiconductor, or more. It aims to describe the distribution of th ...
: \varepsilon \nabla^2 \Phi(\mathbf) = -\, \sum_^N q_j n_j^0 \, \exp\left(- \frac \right) - \rho_(\mathbf) . Solutions to this nonlinear equation are known for some simple systems. Solutions for more general systems may be obtained in the high-temperature (weak coupling) limit, q_j \, \Phi(\mathbf) \ll k_ T, by Taylor expanding the exponential: \exp\left(- \frac \right) \approx 1 - \frac. This approximation yields the linearized Poisson–Boltzmann equation \varepsilon \nabla^2 \Phi(\mathbf) = \left(\sum_^N \frac \right)\, \Phi(\mathbf) -\, \sum_^N n_j^0 q_j - \rho_(\mathbf) which also is known as the Debye–Hückel equation:See The second term on the right-hand side vanishes for systems that are electrically neutral. The term in parentheses divided by \varepsilon, has the units of an inverse length squared and by dimensional analysis leads to the definition of the characteristic length scale \lambda_ = \left(\frac\right)^ that commonly is referred to as the Debye–Hückel length. As the only characteristic length scale in the Debye–Hückel equation, \lambda_D sets the scale for variations in the potential and in the concentrations of charged species. All charged species contribute to the Debye–Hückel length in the same way, regardless of the sign of their charges. For an electrically neutral system, the Poisson equation becomes \nabla^2 \Phi(\mathbf) = \lambda_^ \Phi(\mathbf) - \frac To illustrate Debye screening, the potential produced by an external point charge \rho_ = Q\delta(\mathbf) is \Phi(\mathbf) = \frac e^ The bare Coulomb potential is exponentially screened by the medium, over a distance of the Debye length: this is called Debye screening or shielding (
Screening effect In physics, screening is the damping of electric fields caused by the presence of mobile charge carriers. It is an important part of the behavior of charge-carrying fluids, such as ionized gases (classical plasmas), electrolytes, and charge carr ...
). The Debye–Hückel length may be expressed in terms of the
Bjerrum length The Bjerrum length (after Danish chemist Niels Bjerrum 1879–1958 ) is the separation at which the electrostatic interaction between two elementary charges is comparable in magnitude to the thermal energy scale, k_\text T, where k_\text is the B ...
\lambda_ as \lambda_ = \left(4 \pi \, \lambda_ \, \sum_^N n_j^0 \, z_j^2\right)^, where z_j = q_j/e is the integer
charge number Charge number (''z'') refers to a quantized value of electric charge, with the quantum of electric charge being the elementary charge, so that the charge number equals the electric charge (''q'') in coulombs divided by the elementary-charge cons ...
that relates the charge on the j-th ionic species to the
elementary charge The elementary charge, usually denoted by is the electric charge carried by a single proton or, equivalently, the magnitude of the negative electric charge carried by a single electron, which has charge −1 . This elementary charge is a fundame ...
e.


In a plasma

For a weakly collisional plasma, Debye shielding can be introduced in a very intuitive way by taking into account the granular character of such a plasma. Let us imagine a sphere about one of its electrons, and compare the number of electrons crossing this sphere with and without Coulomb repulsion. With repulsion, this number is smaller. Therefore, according to Gauss theorem, the apparent charge of the first electron is smaller than in the absence of repulsion. The larger the sphere radius, the larger is the number of deflected electrons, and the smaller the apparent charge: this is Debye shielding. Since the global deflection of particles includes the contributions of many other ones, the density of the electrons does not change, at variance with the shielding at work next to a Langmuir probe ( Debye sheath). Ions bring a similar contribution to shielding, because of the attractive Coulombian deflection of charges with opposite signs. This intuitive picture leads to an effective calculation of Debye shielding (see section II.A.2 of ). The assumption of a Boltzmann distribution is not necessary in this calculation: it works for whatever particle distribution function. The calculation also avoids approximating weakly collisional plasmas as continuous media. An N-body calculation reveals that the bare Coulomb acceleration of a particle by another one is modified by a contribution mediated by all other particles, a signature of Debye shielding (see section 8 of ). When starting from random particle positions, the typical time-scale for shielding to set in is the time for a thermal particle to cross a Debye length, i.e. the inverse of the plasma frequency. Therefore in a weakly collisional plasma, collisions play an essential role by bringing a cooperative self-organization process: Debye shielding. This shielding is important to get a finite diffusion coefficient in the calculation of Coulomb scattering (
Coulomb collision A Coulomb collision is a binary elastic collision between two charged particles interacting through their own electric field. As with any inverse-square law, the resulting trajectories of the colliding particles is a hyperbolic Keplerian orbit. This ...
). In a non-isothermic plasma, the temperatures for electrons and heavy species may differ while the background medium may be treated as the vacuum and the Debye length is \lambda_ = \sqrt where * ''λ''D is the Debye length, * ''ε''0 is the
permittivity of free space Vacuum permittivity, commonly denoted (pronounced "epsilon nought" or "epsilon zero"), is the value of the absolute dielectric permittivity of classical vacuum. It may also be referred to as the permittivity of free space, the electric const ...
, * ''k''B is the
Boltzmann constant The Boltzmann constant ( or ) is the proportionality factor that relates the average relative kinetic energy of particles in a gas with the thermodynamic temperature of the gas. It occurs in the definitions of the kelvin and the gas constan ...
, * ''q''''e'' is the
charge of an electron The elementary charge, usually denoted by is the electric charge carried by a single proton or, equivalently, the magnitude of the negative electric charge carried by a single electron, which has charge −1 . This elementary charge is a fundame ...
, * ''Te'' and ''Ti'' are the temperatures of the electrons and ions, respectively, * ''ne'' is the density of electrons, * ''nj'' is the density of atomic species ''j'', with positive
ion An ion () is an atom or molecule with a net electrical charge. The charge of an electron is considered to be negative by convention and this charge is equal and opposite to the charge of a proton, which is considered to be positive by conv ...
ic charge ''zjqe'' Even in quasineutral cold plasma, where ion contribution virtually seems to be larger due to lower ion temperature, the ion term is actually often dropped, giving \lambda_ = \sqrt although this is only valid when the mobility of ions is negligible compared to the process's timescale.


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 the table here below:


In an electrolyte solution

In an electrolyte or a
colloidal suspension A colloid is a mixture in which one substance consisting of microscopically dispersed insoluble particles is suspended throughout another substance. Some definitions specify that the particles must be dispersed in a liquid, while others exte ...
, the Debye lengthInternational Standard ISO 13099-1, 2012, "Colloidal systems – Methods for Zeta potential determination- Part 1: Electroacoustic and Electrokinetic phenomena" for a monovalent electrolyte is usually denoted with symbol ''κ''−1 \kappa^ = \sqrt where * ''I'' is the
ionic strength The ionic strength of a solution is a measure of the concentration of ions in that solution. Ionic compounds, when dissolved in water, dissociate into ions. The total electrolyte concentration in solution will affect important properties such a ...
of the electrolyte in number/m3 units, * ''ε''0 is the
permittivity of free space Vacuum permittivity, commonly denoted (pronounced "epsilon nought" or "epsilon zero"), is the value of the absolute dielectric permittivity of classical vacuum. It may also be referred to as the permittivity of free space, the electric const ...
, * ''ε''r is the dielectric constant, * ''k''B is the
Boltzmann constant The Boltzmann constant ( or ) is the proportionality factor that relates the average relative kinetic energy of particles in a gas with the thermodynamic temperature of the gas. It occurs in the definitions of the kelvin and the gas constan ...
, * ''T'' is the absolute temperature in kelvins, * e is the
elementary charge The elementary charge, usually denoted by is the electric charge carried by a single proton or, equivalently, the magnitude of the negative electric charge carried by a single electron, which has charge −1 . This elementary charge is a fundame ...
, or, for a symmetric monovalent electrolyte, \kappa^ = \sqrt where * ''R'' is the gas constant, * ''F'' is the Faraday constant, * ''C''0 is the electrolyte concentration in molar units (M or mol/L). Alternatively, \kappa^ = \frac where \lambda_ is the
Bjerrum length The Bjerrum length (after Danish chemist Niels Bjerrum 1879–1958 ) is the separation at which the electrostatic interaction between two elementary charges is comparable in magnitude to the thermal energy scale, k_\text T, where k_\text is the B ...
of the medium in nm, and the factor 10^ derives from transforming unit volume from cubic dm to cubic nm. For deionized water at room temperature, at pH=7, ''λ''B ≈ 1μm. At room temperature (), one can consider in water the relation: \kappa^(\mathrm) = \frac where * ''κ''−1 is expressed in nanometres (nm) * ''I'' is the
ionic strength The ionic strength of a solution is a measure of the concentration of ions in that solution. Ionic compounds, when dissolved in water, dissociate into ions. The total electrolyte concentration in solution will affect important properties such a ...
expressed in molar (M or mol/L) There is a method of estimating an approximate value of the Debye length in liquids using conductivity, which is described in ISO Standard, and the book.


In semiconductors

The Debye length has become increasingly significant in the modeling of solid state devices as improvements in lithographic technologies have enabled smaller geometries. The Debye length of semiconductors is given: L_ = \sqrt where * ''ε'' is the dielectric constant, * ''k''B is the Boltzmann constant, * ''T'' is the absolute temperature in kelvins, * ''q'' is the elementary charge, and * ''N''dop is the net density of dopants (either donors or acceptors). When doping profiles exceed the Debye length, majority carriers no longer behave according to the distribution of the dopants. Instead, a measure of the profile of the doping gradients provides an "effective" profile that better matches the profile of the majority carrier density. In the context of solids, Thomas–Fermi screening length may be required instead of Debye length.


See also

*
Bjerrum length The Bjerrum length (after Danish chemist Niels Bjerrum 1879–1958 ) is the separation at which the electrostatic interaction between two elementary charges is comparable in magnitude to the thermal energy scale, k_\text T, where k_\text is the B ...
* Debye–Falkenhagen effect *
Plasma oscillation Plasma oscillations, also known as Langmuir waves (after Irving Langmuir), are rapid oscillations of the electron density in conducting media such as plasmas or metals in the ultraviolet region. The oscillations can be described as an instability ...
*
Shielding effect In chemistry, the shielding effect sometimes referred to as atomic shielding or electron shielding describes the attraction between an electron and the nucleus in any atom with more than one electron. The shielding effect can be defined as a ...
*
Screening effect In physics, screening is the damping of electric fields caused by the presence of mobile charge carriers. It is an important part of the behavior of charge-carrying fluids, such as ionized gases (classical plasmas), electrolytes, and charge carr ...


References


Further reading

* * {{Authority control Electricity Electronics concepts Colloidal chemistry Plasma physics Electrochemistry Length Peter Debye