Plasma parameters

Not to be confused with Plasma parameter.

The complex self-constricting magnetic field lines and current paths in a Birkeland current that may develop in a plasma (Evolution of the Solar System, 1976)

Plasma parameters define various characteristics of a plasma, an electrically conductive collection of charged particles that responds collectively to electromagnetic forces. Plasma typically takes the form of neutral gas-like clouds or charged ion beams, but may also include dust and grains. The behaviour of such particle systems can be studied statistically.

Fundamental plasma parameters

All quantities are in Gaussian (cgs) units except energy and temperature expressed in eV and ion mass expressed in units of the proton mass $\mu =m_{i}/m_{p}$; $Z$ is charge state; $k$ is Boltzmann's constant; $K$ is wavenumber; $\ln \Lambda$ is the Coulomb logarithm.

Frequencies

• electron gyrofrequency, the angular frequency of the circular motion of an electron in the plane perpendicular to the magnetic field:

$\omega _{ce}=eB/m_{e}c=1.76\times 10^{7}B{\mbox{rad/s}}\,$

• ion gyrofrequency, the angular frequency of the circular motion of an ion in the plane perpendicular to the magnetic field:

$\omega _{ci}=ZeB/m_{i}c=9.58\times 10^{3}Z\mu ^{-1}B{\mbox{rad/s}}\,$

• electron plasma frequency, the frequency with which electrons oscillate (plasma oscillation):

$\omega _{pe}=(4\pi n_{e}e^{2}/m_{e})^{1/2}=5.64\times 10^{4}n_{e}^{1/2}{\mbox{rad/s}}$

• ion plasma frequency:

$\omega _{pi}=(4\pi n_{i}Z^{2}e^{2}/m_{i})^{1/2}=1.32\times 10^{3}Z\mu ^{-1/2}n_{i}^{1/2}{\mbox{rad/s}}$

• electron trapping rate:

$\nu _{Te}=(eKE/m_{e})^{1/2}=7.26\times 10^{8}K^{1/2}E^{1/2}{\mbox{s}}^{-1}\,$

• ion trapping rate:

$\nu _{Ti}=(ZeKE/m_{i})^{1/2}=1.69\times 10^{7}Z^{1/2}K^{1/2}E^{1/2}\mu ^{-1/2}{\mbox{s}}^{-1}\,$

• electron collision rate in completely ionized plasmas:

$\nu _{e}=2.91\times 10^{-6}n_{e}\,\ln \Lambda \,T_{e}^{-3/2}{\mbox{s}}^{-1}$

• ion collision rate in completely ionized plasmas:

$\nu _{i}=4.80\times 10^{-8}Z^{4}\mu ^{-1/2}n_{i}\,\ln \Lambda \,T_{i}^{-3/2}{\mbox{s}}^{-1}$

• electron (ion) collision rate in slightly ionized plasmas:

$\nu _{e,i}=N{\overline {\sigma _{e,i}v}}=N\int \limits _{0}^{\infty }\sigma (v)_{e,i}f(v)vdv$

where $\sigma (v)_{e,i}$ is a collision crossection of the electron (ion) on the operating gas atoms (molecules), $f(v)$ is the electron (ion) distribution function in plasma, and $N$ is an operating gas concentration.

Lengths

• Electron thermal de Broglie wavelength, approximate average de Broglie wavelength of electrons in a plasma:

$\Lambda _{e}={\sqrt {\frac {h^{2}}{2\pi m_{e}kT_{e}}}}=6.919\times 10^{-8}\,T_{e}^{-1/2}\,{\mbox{cm}}$

• classical distance of closest approach, the closest that two particles with the elementary charge come to each other if they approach head-on and each have a velocity typical of the temperature, ignoring quantum-mechanical effects:

$e^{2}/kT=1.44\times 10^{-7}\,T^{-1}\,{\mbox{cm}}$

• electron gyroradius, the radius of the circular motion of an electron in the plane perpendicular to the magnetic field:

$r_{e}=v_{Te}/\omega _{ce}=2.38\,T_{e}^{1/2}B^{-1}\,{\mbox{cm}}$

• ion gyroradius, the radius of the circular motion of an ion in the plane perpendicular to the magnetic field:

$r_{i}=v_{Ti}/\omega _{ci}=1.02\times 10^{2}\,\mu ^{1/2}Z^{-1}T_{i}^{1/2}B^{-1}\,{\mbox{cm}}$

• plasma skin depth, the depth in a plasma to which electromagnetic radiation can penetrate:

$c/\omega _{pe}=5.31\times 10^{5}\,n_{e}^{-1/2}\,{\mbox{cm}}$

• Debye length, the scale over which electric fields are screened out by a redistribution of the electrons:

$\lambda _{D}=(kT/4\pi ne^{2})^{1/2}=7.43\times 10^{2}\,T^{1/2}n^{-1/2}\,{\mbox{cm}}$

• Ion inertial length, the scale at which ions decouple from electrons and the magnetic field becomes frozen into the electron fluid rather than the bulk plasma:

$d_{i}=c/\omega _{pi}=2.28\times 10^{7}Z^{-1}(\mu /n_{i})^{1/2}\,{\mbox{cm}}$

• Free path is the average distance between two subsequent collisions of the electron (ion) with plasma components:

$\lambda _{e,i}={\frac {\overline {v_{e,i}}}{\nu _{e,i}}}$

where ${\overline {v_{e,i}}}$ is an average velocity of the electron (ion), and $\nu _{e,i}$ is the electron or ion collision rate.

Velocities

• electron thermal velocity, typical velocity of an electron in a Maxwell–Boltzmann distribution:

$v_{Te}=(kT_{e}/m_{e})^{1/2}=4.19\times 10^{7}\,T_{e}^{1/2}\,{\mbox{cm/s}}$

• ion thermal velocity, typical velocity of an ion in a Maxwell–Boltzmann distribution:

$v_{Ti}=(kT_{i}/m_{i})^{1/2}=9.79\times 10^{5}\,\mu ^{-1/2}T_{i}^{1/2}\,{\mbox{cm/s}}$

• ion sound velocity, the speed of the longitudinal waves resulting from the mass of the ions and the pressure of the electrons:

$c_{s}=(\gamma ZkT_{e}/m_{i})^{1/2}=9.79\times 10^{5}\,(\gamma ZT_{e}/\mu )^{1/2}\,{\mbox{cm/s}}$,

where $\gamma =1+2/n$ is the adiabatic index, and here $n$ is the number of degrees of freedom

• Alfvén velocity, the speed of the waves resulting from the mass of the ions and the restoring force of the magnetic field:

$v_{A}=B/(4\pi n_{i}m_{i})^{1/2}=2.18\times 10^{11}\,\mu ^{-1/2}n_{i}^{-1/2}B\,{\mbox{cm/s}}$

Dimensionless

A 'sun in a test tube'. The Farnsworth-Hirsch Fusor during operation in so called "star mode" characterized by "rays" of glowing plasma which appear to emanate from the gaps in the inner grid.

• square root of electron/proton mass ratio

$(m_{e}/m_{p})^{1/2}=2.33\times 10^{-2}=1/42.9\,$

• number of particles in a Debye sphere

$(4\pi /3)n\lambda _{D}^{3}=1.72\times 10^{9}\,T^{3/2}n^{-1/2}$

• Alfvén velocity/speed of light

$v_{A}/c=7.28\,\mu ^{-1/2}n_{i}^{-1/2}B$

• electron plasma/gyrofrequency ratio

$\omega _{pe}/\omega _{ce}=3.21\times 10^{-3}\,n_{e}^{1/2}B^{-1}$

• ion plasma/gyrofrequency ratio

$\omega _{pi}/\omega _{ci}=0.137\,\mu ^{1/2}n_{i}^{1/2}B^{-1}$

• thermal/magnetic pressure ratio ("beta")

$\beta =8\pi nkT/B^{2}=4.03\times 10^{-11}\,nTB^{-2}$

• magnetic/ion rest energy ratio

$B^{2}/8\pi n_{i}m_{i}c^{2}=26.5\,\mu ^{-1}n_{i}^{-1}B^{2}$

• Coulomb logarithm is an average coefficient taking into account far Coulomb interactions of charged particles in plasma. Its value is evaluated in the nonrelativistic case approximately

for electrons $\ln \Lambda \simeq 13.6$,

for ions $\ln \Lambda \simeq 6.8$

See also

• List of plasma (physics) articles

References

• NRL Plasma Formulary (esp. p. 28 and p. 29), J.D. Huba, Naval Research Laboratory (2007)