# Plasma parameters

(Redirected from Collisionality)

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

### Lengths

$\lambda _{\mathrm {th} ,e}={\sqrt {\frac {h^{2}}{2\pi m_{e}kT_{e}}}}=6.919\times 10^{-8}\,T_{e}^{-1/2}\,{\mbox{cm}}$
• 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 has 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 (also called the electron inertial length), 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:
$\lambda _{D}=(kT_{e}/4\pi ne^{2})^{1/2}=7.43\times 10^{2}\,T_{e}^{1/2}n^{-1/2}\,{\mbox{cm}}$
• 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:
$d_{i}=c/\omega _{pi}=2.28\times 10^{7}\,Z^{-1}(\mu /n_{i})^{1/2}\,{\mbox{cm}}$
• mean free path, 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

$v_{Te}=(kT_{e}/m_{e})^{1/2}=4.19\times 10^{7}\,T_{e}^{1/2}\,{\mbox{cm/s}}$
$v_{Ti}=(kT_{i}/m_{i})^{1/2}=9.79\times 10^{5}\,\mu ^{-1/2}T_{i}^{1/2}\,{\mbox{cm/s}}$
• ion speed of sound, 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 $\gamma$  is the adiabatic index
• 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.
• 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
$B^{2}/8\pi n_{i}m_{i}c^{2}=26.5\,\mu ^{-1}n_{i}^{-1}B^{2}$

## Collisionality

In the study of tokamaks, collisionality is a dimensionless parameter which expresses the ratio of the electron-ion collision frequency to the banana orbit frequency.

The plasma collisionality $\nu ^{*}$  is defined as

$\nu ^{*}=\nu _{\mathrm {ei} }\,{\sqrt {\frac {m_{\mathrm {e} }}{k_{\mathrm {B} }T_{\mathrm {e} }}}}\,\epsilon ^{-3/2}\,qR,$

where $\nu _{\mathrm {ei} }$  denotes the electron-ion collision frequency, $R$  is the major radius of the plasma, $\epsilon$  is the inverse aspect-ratio, and $q$  is the safety factor. The plasma parameters $m_{\mathrm {i} }$  and $T_{\mathrm {i} }$  denote, respectively, the mass and temperature of the ions, and $k_{\mathrm {B} }$  is the Boltzmann constant.