In a plasma, the Boltzmann relation describes the

number density The number density (symbol: ''n'' or ''ρ''N) is an intensive quantity used to describe the degree of concentration of countable objects (particles, molecules, phonons, cells, galaxies, etc.) in physical space: three-dimensional volumetric num ...

of an

isothermal In thermodynamics, an isothermal process is a type of thermodynamic process in which the temperature ''T'' of a system remains constant: Δ''T'' = 0. This typically occurs when a system is in contact with an outside thermal reservoir, an ...

charged particle In physics, a charged particle is a particle with an electric charge. It may be an ion, such as a molecule or atom with a surplus or deficit of electrons relative to protons. It can also be an electron or a proton, or another elementary pa ...

fluid In physics, a fluid is a liquid, gas, or other material that continuously deforms (''flows'') under an applied shear stress, or external force. They have zero shear modulus, or, in simpler terms, are substances which cannot resist any shear ...

when the thermal and the electrostatic forces acting on the fluid have reached equilibrium. In many situations, the electron density of a plasma is assumed to behave according to the Boltzmann relation, due to their small mass and high mobility.

Equation

If the local

electrostatic potential Electrostatics is a branch of physics that studies electric charges at rest ( static electricity). Since classical times, it has been known that some materials, such as amber, attract lightweight particles after rubbing. The Greek word for a ...

s at two nearby locations are ''φ''₁ and ''φ''₂, the Boltzmann relation for the electrons takes the form: :

n_\text (\phi_2) = n_\text(\phi_1) e^

where ''n''_e is the electron

, ''T''_e is the

temperature Temperature is a physical quantity that expresses quantitatively the perceptions of hotness and coldness. Temperature is measured with a thermometer. Thermometers are calibrated in various temperature scales that historically have relied o ...

of the plasma, and ''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 ...

Derivation

A simple derivation of the Boltzmann relation for the electrons can be obtained using the momentum fluid equation of the two-fluid model of

plasma physics Plasma ()πλάσμα
, Henry George Liddell, R ...

in absence of a

magnetic field A magnetic field is a vector field that describes the magnetic influence on moving electric charges, electric currents, and magnetic materials. A moving charge in a magnetic field experiences a force perpendicular to its own velocity and to ...

. When the electrons reach

dynamic equilibrium In chemistry, a dynamic equilibrium exists once a reversible reaction occurs. Substances transition between the reactants and products at equal rates, meaning there is no net change. Reactants and products are formed at such a rate that the co ...

, the inertial and the collisional terms of the momentum equations are zero, and the only terms left in the equation are the pressure and electric terms. For an isothermal fluid, the

pressure Pressure (symbol: ''p'' or ''P'') is the force applied perpendicular to the surface of an object per unit area over which that force is distributed. Gauge pressure (also spelled ''gage'' pressure)The preferred spelling varies by country a ...

force takes the form :

F_=-k_\textT_\text\nabla n_\text,

while the electric term is :

F_=e n_\text \nabla\phi

. Integration leads to the expression given above. In many problems of plasma physics, it is not useful to calculate the electric potential on the basis of the

Poisson 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 ...

because the electron and ion densities are not known ''a priori'', and if they were, because of

quasineutrality Plasma ()πλάσμα
, Henry George Liddell, R ...

the net charge density is the small difference of two large quantities, the electron and ion charge densities. If the electron density is known and the assumptions hold sufficiently well, the electric potential can be calculated simply from the Boltzmann relation.

Inaccurate situations

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 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 ...

s) or when the electrons are prevented from moving by a magnetic field (see e.g. lower hybrid oscillations).

References

* {{Reflist Plasma physics

Equation

Derivation

Inaccurate situations

See also

References