A double layer is a structure

Structure

Structure is a fundamental, tangible or intangible notion referring to the recognition, observation, nature, and permanence of patterns and relationships of entities. This notion may itself be an object, such as a built structure, or an attribute, such as the structure of society...

 in a plasma

Plasma (physics)

In physics and chemistry, plasma is a state of matter similar to gas in which a certain portion of the particles are ionized. Heating a gas may ionize its molecules or atoms , thus turning it into a plasma, which contains charged particles: positive ions and negative electrons or ions...

 and consists of two parallel layers with opposite electrical charge. The sheets of charge cause a strong electric field and a correspondingly sharp change in voltage

Voltage

Voltage, otherwise known as electrical potential difference or electric tension is the difference in electric potential between two points — or the difference in electric potential energy per unit charge between two points...

 (electrical potential) across the double layer. Ions and electrons which enter the double layer are accelerated, decelerated, or reflected by the electric field. In general, double layers (which may be curved rather than flat) separate regions of plasma with quite different characteristics. Double layers are found in a wide variety of plasmas, from discharge tubes to space plasmas to the Birkeland current

Birkeland current

A Birkeland current is a set of currents which flow along geomagnetic field line connecting the Earth’s magnetosphere to the Earth's high latitude ionosphere. They are a specific class of magnetic field-aligned currents. Lately, the term Birkeland currents has been expanded by some authors to...

s supplying the Earth's aurora

Aurora (astronomy)

An aurora is a natural light display in the sky particularly in the high latitude regions, caused by the collision of energetic charged particles with atoms in the high altitude atmosphere...

, and are especially common in current-carrying plasmas. Compared to the sizes of the plasmas which contain them, double layers are very thin (typically ten Debye length

Debye length

In plasma physics, the Debye length , named after the Dutch physicist and physical chemist Peter Debye, is the scale over which mobile charge carriers screen out electric fields in plasmas and other conductors. In other words, the Debye length is the distance over which significant charge...

s), with widths ranging from a few millimeters for laboratory plasmas to thousands of kilometres for astrophysical plasmas.

Other names for a double layer are electrostatic double layer, electric double layer, plasma double layers, electrostatic shock (a type of double layer which is oriented at an oblique angle to the magnetic field in such a way that the perpendicular electric field is much stronger than the parallel electric field), space charge layer. In laser physics, a double layer is sometimes called an ambipolar electric field. Double layers are conceptually related to the concept of a 'sheath' (see Debye sheath

Debye sheath

The Debye sheath is a layer in a plasma which has a greater density of positive ions, and hence an overall excess positive charge, that balances an opposite negative charge on the surface of a material with which it is in contact...

).

The adopted electrical symbol for a double layer, when represented in an electrical circuit is
────DL────. If there is a net current present, then the DL is oriented with the base of the L in line with direction of current.

An overview of double layers in space, experiment and simulation is given in the introduction of ref.

Double layer classification

Double layers may be classified in the following ways:

• Weak and strong double layers. The strength of a double layer is expressed as the ratio of the potential drop in comparison with the plasma’s equivalent thermal energy
  Thermal energy
  Thermal energy is the part of the total internal energy of a thermodynamic system or sample of matter that results in the system's temperature....
  
  , or in comparison with the rest mass energy of the electron
  Electron
  The electron is a subatomic particle with a negative elementary electric charge. It has no known components or substructure; in other words, it is generally thought to be an elementary particle. An electron has a mass that is approximately 1/1836 that of the proton...
  
  s. A double layer is said to be strong if the potential drop across the layer is greater than the equivalent thermal energy of the plasma’s components. This means that for strong double layers there are four different components to the plasma:

1. The electrons entering at the low potential side of the double layer which are accelerated;
2. The ions entering at the high potential side of the double layer which are accelerated;
3. The electrons entering at the high potential side of the double layer which are decelerated and successively reflected; and
4. The ions which enter the double layer at the low potential side of the double layer which are decelerated and reflected.

Note that in the case of a weak double layer, the electrons and ions entering from the “wrong” side are decelerated by the electric field, however most will not be reflected, as the potential drop is not strong enough.

• Relativistic or nonrelativistic double layers. A double layer is said to be relativistic if the potential drop over the layer is so large that the total gain in energy of the particles is larger than the rest mass energy of the electron. The charge distribution in a relativistic double layer is such that the charge density is located in two very thin layers, and inside the double layer the density is constant at and very low compared to the rest of the plasma. In this respect, the double layer is similar to the charge distribution in a capacitor
  Capacitor
  A capacitor is a passive two-terminal electrical component used to store energy in an electric field. The forms of practical capacitors vary widely, but all contain at least two electrical conductors separated by a dielectric ; for example, one common construction consists of metal foils separated...
  
  . As a special case of a relativistic double layer one can consider the vacuum gap at the magnetic polar cap of a pulsar.
• Current carrying and current-free double layers may both occur. Current carrying double layers may be generated by current-driven plasma instabilities which amplify variations of the plasma density. Current-free double layers form on the interface between two plasma regions with different characteristics, and their associated electric field maintains a balance between the penetration of electrons in either direction (so that the net current is low).

Double layer formation

There are two different kinds of double layers, which are formed differently:

Current carrying double layers

Current carrying double layers may arise in plasmas carrying a current. Various instabilities

Instability

In numerous fields of study, the component of instability within a system is generally characterized by some of the outputs or internal states growing without bounds...

 can be responsible for the formation of these layers. One example is the Buneman instability which occurs when the streaming velocity of the electrons (basically the current density divided by the electron density) exceeds the electron thermal velocity

Thermal velocity

The thermal velocity or thermal speed is a typical velocity of the thermal motion of particles which make up a gas, liquid, etc. Thus, indirectly, thermal velocity is a measure of temperature. Technically speaking it is a measure of the width of the peak in the Maxwell-Boltzmann particle velocity...

 of the plasma. Double layers (and other phase space structures) are often formed in the non-linear phase of the instability. One way of viewing the Buneman instability is to describe what happens when the current (in the form of a zero temperature electron beam) has to pass through a region of decreased ion density. In order to prevent charge from accumulating, the current in the system must be the same everywhere (in this 1D model). The electron density also has to be close to the ion density (quasineutrality), so there is also a dip in electron density. The electrons must therefore be accelerated into the density cavity, to maintain the same current density with a lower density of charge carriers. This implies that the density cavity is at a high electrical potential. As a consequence, the ions are accelerated out of the cavity, amplifying the density perturbation. Then there is the situation of a double-double layer, of which one side will most likely be convected away by the plasma, leaving a regular double layer. This is the process in which double layers are produced along planetary magnetic field lines in so-called Birkeland current

Birkeland current

A Birkeland current is a set of currents which flow along geomagnetic field line connecting the Earth’s magnetosphere to the Earth's high latitude ionosphere. They are a specific class of magnetic field-aligned currents. Lately, the term Birkeland currents has been expanded by some authors to...

s.

Current-free double layers

Current-free double layers occur at the boundary between plasma regions with different plasma properties. We explain how they form (neglecting the ions which are considered solely as a neutralizing background). Consider a plasma divided into two regions by a plane, which has a higher electron temperature on one side than on the other (the same analysis can also be done for different densities). This means that the electrons on one side of the interface have a greater thermal velocity. The electrons may stream freely in either direction, and the flux of electrons from the hot plasma to the cold plasma will be greater than the flux of the electrons from the cold plasma to the hot plasma, because the electrons from the hot side have a greater average speed. Because many more electrons enter the cold plasma than exit it, part of the cold region becomes negatively charged. The hot plasma, conversely, becomes positively charged. Therefore, an electric field builds up, which starts to accelerate electrons towards the hot region, reducing the net flux. In the end, the electric field builds up until the fluxes of electrons in either direction are equal, and further charge build up in the two plasmas is prevented. The potential drop is in fact exactly equal to the difference in thermal energy

Thermal energy

Thermal energy is the part of the total internal energy of a thermodynamic system or sample of matter that results in the system's temperature....

 between the two plasma regions in this case, so such a double layer is a marginally strong double layer.

Double layer formation mechanisms

The details of the formation mechanism depend on the environment of the plasma (e.g. double layers in the laboratory, ionosphere, solar wind, fusion, etc.). Proposed mechanisms for their formation have included:

• 1971: Between plasmas of different temperatures
• 1976: In laboratory plasmas
• 1982: Disruption of a neutral current sheet
  Current sheet
  A current sheet is an electric current that is confined to a surface, rather than being spread through a volume of space. Current sheets feature in magnetohydrodynamics , the study of the behavior of electrically conductive fluids: if there is an electric current through part of the volume of such...
  
  
• 1983: Injection of non-neutral electron current into a cold plasma
• 1985: Increasing the current density in a plasma
• 1986: In the accretion column of a neutron star
• 1986: By pinches in cosmic plasma regions
• 1987: In a plasma constrained by a magnetic mirror
• 1988: By an electrical discharge
• 1988: Current-driven instabilities (strong double layers)
• 1988: Spacecraft-ejected electron beams
• 1989: From shock waves in a plasma
• 2000: Laser radiation
• 2002: When magnetic field-aligned currents encounter density cavities
• 2003: By the incidence of plasma on the dark side of the Moon's surface

Features and characteristics of double layers

• Thickness: The production of a double layer requires regions with a significant excess of positive or negative charge, that is, where quasi-neutrality is violated. In general, quasi-neutrality can only be violated on scales of the order of the Debye length
  Debye length
  In plasma physics, the Debye length , named after the Dutch physicist and physical chemist Peter Debye, is the scale over which mobile charge carriers screen out electric fields in plasmas and other conductors. In other words, the Debye length is the distance over which significant charge...
  
  . The thickness of a double layer is of the order of ten Debye lengths, which is a few centimeters in the ionosphere
  Ionosphere
  The ionosphere is a part of the upper atmosphere, comprising portions of the mesosphere, thermosphere and exosphere, distinguished because it is ionized by solar radiation. It plays an important part in atmospheric electricity and forms the inner edge of the magnetosphere...
  
  , a few tens of meters in the interplanetary medium
  Interplanetary medium
  The interplanetary medium is the material which fills the solar system and through which all the larger solar system bodies such as planets, asteroids and comets move.-Composition and physical characteristics:...
  
  , and tens of kilometers in the intergalactic medium.
• Particle acceleration: The potential drop across the double layer will accelerate electrons and positive ions in opposite directions. The magnitude of the potential drop determines the acceleration of the charged particles. In strong double layers, this will result in beams or jets of charged particles.
• Particle populations: As described in the formation of double layers, there are four populations of charge particles inside a double layer. Note that in the case of weak double layers not all electrons and ions entering "from the wrong side" will be reflected, and therefore there will also be a population of decelerated electrons and ions.
• Particle flux: For non-relativistic current carrying double layers the electrons comprise the main part of the particle flux. The Langmuir condition states that the ratio of the electron and the ion current across the layer is given by the square root of the mass ratio of the ions to the electrons. For relativistic double layers the current ratio is 1; i.e. equal amounts of current are carried by the electrons and the ions.
• Energy supply: In a certain limit, the voltage drop across a current-carrying double layer is proportional to the total current, and it might be thought of as a resistive
  Resistor
  A linear resistor is a linear, passive two-terminal electrical component that implements electrical resistance as a circuit element.The current through a resistor is in direct proportion to the voltage across the resistor's terminals. Thus, the ratio of the voltage applied across a resistor's...
  
   element (or load) which absorbs energy in an electric circuit. Anthony Peratt (1991) wrote: "Since the double layer acts as a load, there has to be an external source maintaining the potential difference and driving the current. In the laboratory this source is usually an electrical power supply, whereas in space it may be the magnetic energy stored in an extended current system, which responds to a change in current with an inductive voltage".
• Stability: Double layers in laboratory plasmas may be stable or unstable depending on the parameter regime. Various types of instabilities may occur, often arising due to the formation of beam
  Charged particle beam
  A charged particle beam is a spatially localized group of electrically charged particles that have approximately the same velocity . The kinetic energies of the particles are typically measured in keV or MeV, much larger than the energies of particles at ambient temperature...
  
  s of ions and electrons. Unstable double layers are noisy in the sense that they produce oscillations across a wide frequency band. A lack of plasma stability may also lead to a dramatic change in configuration often referred to as an explosion
  Explosion
  An explosion is a rapid increase in volume and release of energy in an extreme manner, usually with the generation of high temperatures and the release of gases. An explosion creates a shock wave. If the shock wave is a supersonic detonation, then the source of the blast is called a "high explosive"...
  
   (and hence exploding double layer). In one example, the region enclosed in the double layer rapidly expands and evolves. An explosion
  Explosion
  An explosion is a rapid increase in volume and release of energy in an extreme manner, usually with the generation of high temperatures and the release of gases. An explosion creates a shock wave. If the shock wave is a supersonic detonation, then the source of the blast is called a "high explosive"...
  
   of this type was first discovered in mercury arc rectifiers used in high-power direct-current transmission lines, where the voltage drop across the device was seen to increase by several orders of magnitude. Double layers may also drift, usually in the direction of the emitted electron beam, and in this respect are natural analogues to the smooth—bore magnetron.) (not to be confused with a unit of magnetic moment, the Bohr magneton, which is created by the "classical circular motion" of an electron around a proton).

Location Typical Voltage drop Source Ionosphere 102–104V Satellite Solar 109–1011V Estimated Neutron star 1015V Estimated Galactic filament 1017V Estimated Typical Double Layers

• Magnetised plasmas: Double layers can both form in normal and magnetized plasmas.
• Cellular nature: While double layers are relatively thin, they will spread over the entire cross surface of a laboratory container. Likewise where adjacent plasma regions have different properties, double layers will form and tend to cellularise the different regions.
• Energy transfer: Double layers facilitate the transfer of electrical energy into kinetic energy, dW/dt=I.ΔV where I is the electric current dissipating energy into a double layer with a voltage drop of ΔV. Alfvén points out that the current may well consist exclusively of low-energy particles. Torvén et al. also report that plasma may spontaneously transfer magnetically stored energy into kinetic energy by electric double layers.
• Oblique double layer: An oblique double layer has its electric field not parallel to the background magnetic field; i.e., it is not field-aligned.
• Simulation: Double layers may be modelled using kinetic computer models like particle-in-cell (PIC) simulations. In some cases it is reasonable to treat the plasma as essentially one- or two-dimensional to reduce the computational cost of a simulation.
• Bohm Criterion: A double layer cannot exist under all circumstances. In order to achieve that the electric field vanishes at the boundaries of the double layer, an existence criterion says that there is a maximum to the temperature of the ambient plasma. This is the so-called Bohm criterion. A mathematical description is given in the math section. In the theory of the Debye sheath
  Debye sheath
  The Debye sheath is a layer in a plasma which has a greater density of positive ions, and hence an overall excess positive charge, that balances an opposite negative charge on the surface of a material with which it is in contact...
  
   there is a related but not identical condition also known as the Bohm criterion.
• Bio-physical analogy: A model of plasma double layers has been used to investigate their applicability to understanding ion transport across biological cell membranes. Brazilian researchers have note that "Concepts like charge neutrality, Debye length
  Debye length
  In plasma physics, the Debye length , named after the Dutch physicist and physical chemist Peter Debye, is the scale over which mobile charge carriers screen out electric fields in plasmas and other conductors. In other words, the Debye length is the distance over which significant charge...
  
  , and double layer are very useful to explain the electrical properties of a cellular membrane." Plasma physicist Hannes Alfvén
  Hannes Alfvén
  Hannes Olof Gösta Alfvén was a Swedish electrical engineer, plasma physicist and winner of the 1970 Nobel Prize in Physics for his work on magnetohydrodynamics . He described the class of MHD waves now known as Alfvén waves...
  
   also noted that association of double layers with cellular structure, as had Irving Langmuir
  Irving Langmuir
  Irving Langmuir was an American chemist and physicist. His most noted publication was the famous 1919 article "The Arrangement of Electrons in Atoms and Molecules" in which, building on Gilbert N. Lewis's cubical atom theory and Walther Kossel's chemical bonding theory, he outlined his...
  
   before him, who coined the named "plasma" after its resemblance to blood cells.

History of double layers

The research of these objects is a relatively young phenomenon. Although it was already known in the 1920s that a plasma has a limited capacity for current maintenance, Irving Langmuir

Irving Langmuir

Irving Langmuir was an American chemist and physicist. His most noted publication was the famous 1919 article "The Arrangement of Electrons in Atoms and Molecules" in which, building on Gilbert N. Lewis's cubical atom theory and Walther Kossel's chemical bonding theory, he outlined his...

 characterized double layers in the laboratory and called these structures double-sheaths. It was not until the 1950s that a thorough study of double layers started in the laboratory (e.g. Schönhuber, 1958). At the moment many groups are working on this topic theoretically, experimentally and numerically.
It was first proposed by Hannes Alfvén

Hannes Alfvén

Hannes Olof Gösta Alfvén was a Swedish electrical engineer, plasma physicist and winner of the 1970 Nobel Prize in Physics for his work on magnetohydrodynamics . He described the class of MHD waves now known as Alfvén waves...

 (the developer of magnetohydrodynamics) that the polar lights or Aurora Borealis are created by accelerated electrons in the magnetosphere of the Earth. He supposed that the electrons were accelerated electrostatically by an electric field localized in a small volume bounded by two charged regions. This so-called double layer would accelerate electrons Earthwards. Many experiments with rockets and satellites have been performed to investigate the magnetosphere and acceleration regions. The first indication for the existence of electric field aligned along the magnetic field (or double layers) in the magnetosphere was by a rocket experiment by McIlwain (1960). Later, in 1977, Forrest Mozer reported that satellites had detected the signature of double layers (which he called electrostatic shocks) in the magnetosphere.

The most definite proof of these structures was obtained by the Viking satellite, which measures the differential potential structures in the magnetosphere with probes mounted on 40m long booms. These probes can measure the local particle density and the potential difference between two points 80m apart. Asymmetric potential structures with respect to 0 V were measured, which means that the structure has a net potential and can be regarded as a double layer. The particle densities measured in such structures can be as low as 33% of the background density. The structures usually have an extent of 100 m (a few tens of Debye lengths). The filling factor of the lower magnetosphere with such solitary structures is about 10%. If one out of 5 such structures has a net potential drop of 1 V then the total potential drop over a region of 5000 km would be more than the 1 kV which is needed for the electrons to create the aurora. Magnetospheric double layers typically have a strength (where the electron temperature is assumed to lie in the range () and are therefore weak.
The American FAST

Fast Auroral Snapshot Explorer

The Fast Auroral Snapshot Explorer was launched from Vandenberg Air Force Base on board a Pegasus XL rocket on August 21, 1996. One in the series of NASA's Small Explorer spacecraft, FAST was designed to observe and measure the plasma physics of the auroral phenomena which occur around both poles...

 spacecraft found strong double layers in the auroral acceleration region.
Strong Double layers were also found in the return current region by Andersson et al. . The return current region is where electrons move upward from the ionosphere to close the auroral current circuit.

In the laboratory, double layers can be created in different devices. They are investigated in double plasma machines, triple plasma machines, and Q-machine

Q-machine

A Q-machine is a device that is used in experimental plasma physics.The name Q-machine stems from the original intention of creating aquiescent plasma that is free from the fluctuations that arepresent in plasmas created in electric discharges...

s. The stationary potential structures which can be measured in these machines agree very well with what one would expect theoretically. An example of a laboratory double layer can be seen in the figure below, taken from Torvén and Linberg (1980), where we can see how well-defined and confined the potential drop of a double layer in a double plasma machine is.
One of the interesting things of the experiment by Torvén and Lindberg (1980) is that not only did they measure the potential structure in the double plasma machine but they also found high-frequency fluctuating electric fields at the high-potential side of the double layer (also shown in the figure). These fluctuations are probably due to a beam-plasma interaction outside the double layer which excites plasma turbulence. Their observations are consistent with experiments on electromagnetic radiation emitted by double layers in a double plasma machine by Volwerk (1993), who, however, also observed radiation from the double layer itself. The power of these fluctuations has a maximum around the plasma frequency of the ambient plasma.
It was later found that the electrostatic high-frequency fluctuations near the double layer can be concentrated in a narrow region, sometimes called the hf-spike , on the high potential side of the double layer.
Subsequently, both radio emissions, near the plasma frequency, and whistler waves at much lower frequencies were seen to emerge from this region .
Similar whistler wave structures were observed together with electron beams near Saturn's moon Enceladus

Enceladus (moon)

Enceladus is the sixth-largest of the moons of Saturn. It was discovered in 1789 by William Herschel. Until the two Voyager spacecraft passed near it in the early 1980s very little was known about this small moon besides the identification of water ice on its surface...

 , suggesting the presence of a double layer at lower altitude.

A recent development in double layer experiments is the investigation of so-called stairstep double layers. It has been observed that a potential drop in a plasma column can be split up into different parts. Transitions from a single double layer into two-, three-, or greater-step double layers are strongly sensitive to the boundary conditions of the plasma (Hershkowitz, 1992). These experiments can give us information about the formation of the magnetospheric double layers and their possible role in creating the aurora.

Some scientists have subsequently suggested a role of double layers in solar flares.

Mathematical description of a double layer

In this section we will take a closer look at the mathematics behind double layers. We first describe a semi-quantitative criterion for the formation of a density dip. We then describe a particularly simple kind of double layer. We then explain how to use the distribution function

Distribution function

In molecular kinetic theory in physics, a particle's distribution function is a function of seven variables, f, which gives the number of particles per unit volume in phase space. It is the number of particles per unit volume having approximately the velocity near the place and time...

 and the Vlasov-Poisson equation to model more-complex double layers.

Formation of a density dip

First we will take a look at the generation of a double layer in a current-carrying plasma. In 1968 Alfvén and Carlqvist showed that a density dip in a current carrying plasma can be favorable for the generation of a double layer. In this case we look at the plasma as a combination of two fluids, the moving electron fluid and the immobile ion fluid which acts as a neutralizing background. The electron fluid is treated as an essentially zero temperature beam and the ions are assumed to be collisional, and possess some finite temperature.

The density dip in the plasma (of both electrons and ions) will cause an electric field to be generated in order to keep the current density at the same level; i.e., electrons are accelerated in the decreasing part into the dip and decelerated in the increasing part out of the dip. However, this electric field will also have an influence on the first as immobile assumed ions. These ions will be driven out of the density dip, increasing it, and thereby increasing the electric field. When all ions are gone, the electric field has reached its maximum value over the dip. Note that we then have a double-double layer (increasing and decreasing electric field), and one side needs to be transported away.

We will use the quasi-static, non-relativistic description of this mechanism, which is governed by the continuity equation

Continuity equation

A continuity equation in physics is a differential equation that describes the transport of a conserved quantity. Since mass, energy, momentum, electric charge and other natural quantities are conserved under their respective appropriate conditions, a variety of physical phenomena may be described...

 and the momentum

Momentum

In classical mechanics, linear momentum or translational momentum is the product of the mass and velocity of an object...

 equation:


Combining these two equations we get an expression for the electric field dependent on the electron density:

where the electron current density. The ions will experience an outward force due to this electric field, with
.
Only when the force of the electric field can overcome the force by the ion pressure

Pressure

Pressure is the force per unit area applied in a direction perpendicular to the surface of an object. Gauge pressure is the pressure relative to the local atmospheric or ambient pressure.- Definition :...

 gradient

Gradient

In vector calculus, the gradient of a scalar field is a vector field that points in the direction of the greatest rate of increase of the scalar field, and whose magnitude is the greatest rate of change....

 can the evacuation of the density dip take place. Comparing the two forces (pressure and electric) assuming a quasi-neutral, thermal

Thermal

A thermal column is a column of rising air in the lower altitudes of the Earth's atmosphere. Thermals are created by the uneven heating of the Earth's surface from solar radiation, and are an example of convection. The sun warms the ground, which in turn warms the air directly above it...

 plasma shows after integration that this can only happen when . This happens to be the Bohm criterion for the stability of a double layer (more below).

Current-carrying double layers formed by single, zero temperature beams

We consider how a single zero-temperature beam of ions and a single zero-temperature beam of electrons, together with a trapped, zero-velocity ion component, and a trapped, zero-velocity electron component, may form a particular class of double layer. The trapped components are referred to as the 'ambient plasma' and will later be allowed to have finite temperature.

We use Poisson's equation and the conservation of momentum and number density to analyse the structure of these double layers, in the 1D, time-independent limit. We are looking for double-layerlike solutions, where there is a well localised region with a potential gradient, outside of which the electric field is zero. The region can be divided into the interval inside the double layer, where there is only one ion component and one electron component, but there is a finite field, and the outside region, where the electric field is zero. For the moment, we need only consider the inside region and the densities and velocities associated with the beams inside the layer.

The electron beam component is streaming with positive velocity (to the right), and the ion beam is streaming with negative velocity (to the left). Here, the conservation of particle energy means that is a constant, and the conservation of particle number means that the current is also a constant.


where and . Here and are respectively the electron (and ion) density and particle drift speed at the low (high) potential side of the double layer.

Now we use Poisson’s equation to obtain the maximal current through the double layer, as a function of the potential drop, the fraction of current carried by the ions as compared to the electrons and a temperature limit for the ambient plasma. We chose and , with the thickness of the double layer.


Thus we can write Poisson’s equation in the region inside the double layer as


Introducing an integration factor at both sides and integrating over at the left hand side and over on the right hand side the first integration leads to the square of the electric field . The assumption that there is no electric field outside the double layer then leads to the ‘’Langmuir condition’’ for non-relativistic double layers:


For this double layer (in a hydrogen plasma) the electron current dominates the ion current by a factor of . (Note that for the same theory for ultra-relativistic double layers gives this fraction equal to 1). Further integration, as done by Raadu (1989), then leads to the Langmuir-Child relation:


where is expressed in terms of the elliptical integrals E and K:


If we now allow the ambient plasma to be at finite temperature we have to take into account reflected particles more carefully and examine how far they can penetrate into the repulsive electric field. We describe the ambient plasma by a Boltzmann distribution over the double layer:


The densities of the reflected particles are now added to Poisson’s equation. In order that the particles in the 'ambient plasma' be truly trapped we require that their temperature be lower than the double layer potential. This can be seen in terms of the restriction that the potential and the electric field have to vanish at the boundaries of the double layer. The precise condition is known as the ‘’Bohm criterion’’:


A double layer of this type cannot form if this criterion is not met. This is the same condition under which a double layer can be formed by an ion density dip (or equivalently, for instability to parallel wavemodes like the ion acoustic or Buneman instability) as discussed before.

The Vlasov–Poisson equation

In general the plasma distributions near a double layer are necessarily strongly non-Maxwellian, and therefore inaccessible to fluid models. In order to analyse double layers in full generality, the plasma must be described using the particle distribution function

Distribution function

In molecular kinetic theory in physics, a particle's distribution function is a function of seven variables, f, which gives the number of particles per unit volume in phase space. It is the number of particles per unit volume having approximately the velocity near the place and time...

 , which describes the number of particles of species having approximately the velocity

Velocity

In physics, velocity is speed in a given direction. Speed describes only how fast an object is moving, whereas velocity gives both the speed and direction of the object's motion. To have a constant velocity, an object must have a constant speed and motion in a constant direction. Constant ...

near the place and time .

The Vlasov-Poisson equations give the time-dependent interaction of a plasma (described using the particle distribution) with its self-consistent electric field. The Vlasov-Poisson equations are a combination of the Vlasov equation

Vlasov equation

The Vlasov equation is a differential equation describing time evolution of the distribution function of plasma consisting of charged particles with long-range interaction...

 for each species (we take the nonrelativistic zero-magnetic field limit):


and Poisson’s equation for self-consistent electric field:


Here is the particle’s electric charge, is the particle’s mass, is the electric field

Electric field

In physics, an electric field surrounds electrically charged particles and time-varying magnetic fields. The electric field depicts the force exerted on other electrically charged objects by the electrically charged particle the field is surrounding...

, the electric potential

Electric potential

In classical electromagnetism, the electric potential at a point within a defined space is equal to the electric potential energy at that location divided by the charge there...

 and is the electric charge

Electric charge

Electric charge is a physical property of matter that causes it to experience a force when near other electrically charged matter. Electric charge comes in two types, called positive and negative. Two positively charged substances, or objects, experience a mutual repulsive force, as do two...

density.

External links

• A double layer review (1978), L.P. Block
• Electrostatic double layers and a plasma evacuation process (1981), Raadu, M. A.; Carlqvist, P.
• On the physics of relativistic double layers (1982), Per Carlqvist
• On the role of double layers in astrophysical plasmas (1985), Smith, R. A.
• Dynamical aspects of electrostatic double layers (1988) M. A. Raadu, & J. J. Rasmussen
• Filamentary Double Layers (PDF, 1993), W.L. Theisen, R.T.Carpenter, R.L.Merlino
• Energy release in double layers (1985), Raadu, M. A.
• Parallel electric fields accelerating ions and electrons in the same direction (1988), Hultqvist, Bengt; Lundin, Rickard
• Parallel electric fields in the upward current region of the aurora: Numerical solutions (2002, PDF), R.W. Ergun, et al.
• Numerical modeling of low-pressure plasmas: applications to electric double layers (2006, PDF), A. Meige, PhD thesis