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Boltzmann relation
 
 
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In a plasma
Plasma (physics)

In physics and chemistry, plasma is a partially ionized gas, in which a certain proportion of electrons are free rather than being bound to an atom or molecule....
, the Boltzmann relation connects the electron density ne to the plasma potential fpl as follows:

ne = n0 exp(eφpl/kB
Boltzmann constant

The Boltzmann constant is the physical constant relating energy at the particle level with temperature observed at the bulk level. It is the gas constant R divided by the Avogadro constant NA:...
Te)


The reference for the potential is taken to be a position where the electron density is n0. Formally, this is sometimes taken as infinity.

It can be derived in a particle view by equating the density of states to the physical density and applying the Boltzmann factor
Boltzmann factor

In physics, the Boltzmann factor is a weighting factor that determines the relative probability of a state in a multi-state system in thermodynamic equilibrium at temperature ....
.

Alternatively, it can be derived from the fluid equation for the electrons by equating the force density due to the electron pressure gradient assuming isothermal electrons, , to the force density due to the electric field on the electron charge density
Charge density

The linear, surface, or volume charge density is the amount of electric charge in a line , surface, or volume. It is measured in coulombs per metre , square metre , or cubic metre , respectively....
, .

In many problems of plasma physics, it is not useful to calculate the electric potential on the basis of the Poisson equation because the electron and ion densities are not known a priori, and if they were, because of quasineutrality
Plasma (physics)

In physics and chemistry, plasma is a partially ionized gas, in which a certain proportion of electrons are free rather than being bound to an atom or molecule....
 the net charge density is the small difference of two large quantities, the electron and ion charge densities.






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In a plasma
Plasma (physics)

In physics and chemistry, plasma is a partially ionized gas, in which a certain proportion of electrons are free rather than being bound to an atom or molecule....
, the Boltzmann relation connects the electron density ne to the plasma potential fpl as follows:

ne = n0 exp(eφpl/kB
Boltzmann constant

The Boltzmann constant is the physical constant relating energy at the particle level with temperature observed at the bulk level. It is the gas constant R divided by the Avogadro constant NA:...
Te)


The reference for the potential is taken to be a position where the electron density is n0. Formally, this is sometimes taken as infinity.

It can be derived in a particle view by equating the density of states to the physical density and applying the Boltzmann factor
Boltzmann factor

In physics, the Boltzmann factor is a weighting factor that determines the relative probability of a state in a multi-state system in thermodynamic equilibrium at temperature ....
.

Alternatively, it can be derived from the fluid equation for the electrons by equating the force density due to the electron pressure gradient assuming isothermal electrons, , to the force density due to the electric field on the electron charge density
Charge density

The linear, surface, or volume charge density is the amount of electric charge in a line , surface, or volume. It is measured in coulombs per metre , square metre , or cubic metre , respectively....
, .

In many problems of plasma physics, it is not useful to calculate the electric potential on the basis of the Poisson equation because the electron and ion densities are not known a priori, and if they were, because of quasineutrality
Plasma (physics)

In physics and chemistry, plasma is a partially ionized gas, in which a certain proportion of electrons are free rather than being bound to an atom or molecule....
 the net charge density is the small difference of two large quantities, the electron and ion charge densities. If the ion density is known and the assumptions hold sufficiently well, the electric potential can be calculated simply from the Boltzmann relation.

Discrepancies with the Boltzmann relation can occur, for example, when oscillations occur so fast that the electrons cannot find a new equilibrium (see e.g. plasma oscillation
Plasma oscillation

Plasma oscillations, also known as "Langmuir waves" , are rapid oscillations of the electron density in conducting media such as Plasma or metals....
s) or when the electrons are prevented from moving by a magnetic field (see e.g. lower hybrid oscillation
Lower hybrid oscillation

A lower hybrid oscillation is a longitudinal wave oscillation of ions and electrons in a magnetized Plasma . The direction of propagation must be very nearly perpendicular to the stationary magnetic field, within about √ radians....
s).

See also


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