Plasma parameters

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.^[1] The behaviour of such particle systems can be studied statistically.^[2]

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}=1.76\times 10^{11}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 $\ln \Lambda \simeq 6.8$

References

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

Footnotes

↑ Peratt, Anthony, Physics of the Plasma Universe (1992);
↑ Parks, George K., Physics of Space Plasmas (2004, 2nd Ed.)

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