# 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 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:
$\omega_{ce} = eB/m_ec = 1.76 \times 10^7 B \mbox{rad/s} \,$
• ion gyrofrequency, the angular frequency of the circular motion of an ion in the plane perpendicular to the magnetic field:
$\omega_{ci} = ZeB/m_ic = 9.58 \times 10^3 Z \mu^{-1} B \mbox{rad/s} \,$
• electron plasma frequency, the frequency with which electrons oscillate (plasma oscillation):
$\omega_{pe} = (4\pi n_ee^2/m_e)^{1/2} = 5.64 \times 10^4 n_e^{1/2} \mbox{rad/s}$
• ion plasma frequency:
$\omega_{pi} = (4\pi n_iZ^2e^2/m_i)^{1/2} = 1.32 \times 10^3 Z \mu^{-1/2} n_i^{1/2} \mbox{rad/s}$
• 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

$\Lambda_e= \sqrt{\frac{h^2}{2\pi m_ekT_e}}= 6.919\times 10^{-8}\,T_e^{-1/2}\,\mbox{cm}$
• 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:
$e^2/kT=1.44\times10^{-7}\,T^{-1}\,\mbox{cm}$
• 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:
$r_i = v_{Ti}/\omega_{ci} = 1.02\times10^2\,\mu^{1/2}Z^{-1}T_i^{1/2}B^{-1}\,\mbox{cm}$
• plasma skin depth, the depth in a plasma to which electromagnetic radiation can penetrate:
$c/\omega_{pe} = 5.31\times10^5\,n_e^{-1/2}\,\mbox{cm}$
• Debye length, the scale over which electric fields are screened out by a redistribution of the electrons:
$\lambda_D = (kT/4\pi ne^2)^{1/2} = 7.43\times10^2\,T^{1/2}n^{-1/2}\,\mbox{cm}$
• 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:
$d_i = c/\omega_{pi}$
• 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

$v_{Te} = (kT_e/m_e)^{1/2} = 4.19\times10^7\,T_e^{1/2}\,\mbox{cm/s}$
$v_{Ti} = (kT_i/m_i)^{1/2} = 9.79\times10^5\,\mu^{-1/2}T_i^{1/2}\,\mbox{cm/s}$
• ion sound velocity, the speed of the longitudinal waves resulting from the mass of the ions and the pressure of the electrons:
$c_s = (\gamma ZkT_e/m_i)^{1/2} = 9.79\times10^5\,(\gamma ZT_e/\mu)^{1/2}\,\mbox{cm/s}$,

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:
$v_A = B/(4\pi n_im_i)^{1/2} = 2.18\times10^{11}\,\mu^{-1/2}n_i^{-1/2}B\,\mbox{cm/s}$

### 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
$(m_e/m_p)^{1/2} = 2.33\times10^{-2} = 1/42.9 \,$
• number of particles in a Debye sphere
$(4\pi/3)n\lambda_D^3 = 1.72\times10^9\,T^{3/2}n^{-1/2}$
• Alfvén velocity/speed of light
$v_A/c = 7.28\,\mu^{-1/2}n_i^{-1/2}B$
• electron plasma/gyrofrequency ratio
$\omega_{pe}/\omega_{ce} = 3.21\times10^{-3}\,n_e^{1/2}B^{-1}$
• ion plasma/gyrofrequency ratio
$\omega_{pi}/\omega_{ci} = 0.137\,\mu^{1/2}n_i^{1/2}B^{-1}$
• thermal/magnetic pressure ratio ("beta")
$\beta = 8\pi nkT/B^2 = 4.03\times10^{-11}\,nTB^{-2}$
• magnetic/ion rest energy ratio
$B^2/8\pi n_im_ic^2 = 26.5\,\mu^{-1}n_i^{-1}B^2$
• 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 $\ln\Lambda \simeq 6.8$