Plasma parameters

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.

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)

Frequencies

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

\( {\displaystyle \omega _{ce}={\frac {eB}{m_{e}c}}\approx 1.76\times 10^{7}\,B\ {\mbox{rad/s}}\,} \)

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

\( {\displaystyle \omega _{ci}={\frac {ZeB}{m_{i}c}}\approx 9.58\times 10^{3}\,{\frac {ZB}{\mu }}\ {\mbox{rad/s}}\,} \)

electron plasma frequency, the frequency with which electrons oscillate (plasma oscillation):
\( {\displaystyle \omega _{pe}=\left({\frac {4\pi n_{e}e^{2}}{m_{e}}}\right)^{\frac {1}{2}}\approx 5.64\times 10^{4}\,{n_{e}}^{\frac {1}{2}}\ {\mbox{rad/s}}} \)

ion plasma frequency:
\( {\displaystyle \omega _{pi}=\left({\frac {4\pi n_{i}Z^{2}e^{2}}{m_{i}}}\right)^{\frac {1}{2}}\approx 1.32\times 10^{3}\,Z\left({\frac {n_{i}}{\mu }}\right)^{\frac {1}{2}}\ {\mbox{rad/s}}} \)

electron trapping rate:
\( {\displaystyle \nu _{Te}=\left({\frac {eKE}{m_{e}}}\right)^{\frac {1}{2}}\approx 7.26\times 10^{8}\,\left(KE\right)^{\frac {1}{2}}\ /{\mbox{s}}\,} \)

ion trapping rate:
\( {\displaystyle \nu _{Ti}=\left({\frac {ZeKE}{m_{i}}}\right)^{\frac {1}{2}}\approx 1.69\times 10^{7}\,\left({\frac {ZKE}{\mu }}\right)^{\frac {1}{2}}\ /{\mbox{s}}\,} \)

electron collision rate in completely ionized plasmas:
\( {\displaystyle \nu _{e}\approx 2.91\times 10^{-6}\,{\frac {n_{e}\ln \Lambda }{T_{e}^{\frac {3}{2}}}}\ /{\mbox{s}}} \)

ion collision rate in completely ionized plasmas:
\( {\displaystyle \nu _{i}\approx 4.80\times 10^{-8}\,{\frac {Z^{4}n_{i}\,\ln \Lambda }{\left(T_{i}^{3}\mu \right)^{\frac {1}{2}}}}\ /{\mbox{s}}} \)

Lengths

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

\( {\displaystyle \lambda _{\mathrm {th} ,e}={\sqrt {\frac {h^{2}}{2\pi m_{e}kT_{e}}}}\approx 6.919\times 10^{-8}\,{\frac {1}{{T_{e}}^{\frac {1}{2}}}}\ {\mbox{cm}}} \)

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:

\( {\displaystyle {\frac {e^{2}}{kT}}\approx 1.44\times 10^{-7}\,{\frac {1}{T}}\ {\mbox{cm}}} \)

electron gyroradius, the radius of the circular motion of an electron in the plane perpendicular to the magnetic field:
\( {\displaystyle r_{e}={\frac {v_{Te}}{\omega _{ce}}}\approx 2.38\,{\frac {{T_{e}}^{\frac {1}{2}}}{B}}\ {\mbox{cm}}} \)

ion gyroradius, the radius of the circular motion of an ion in the plane perpendicular to the magnetic field:
\( {\displaystyle r_{i}={\frac {v_{Ti}}{\omega _{ci}}}\approx 1.02\times 10^{2}\,{\frac {\left(\mu T_{i}\right)^{\frac {1}{2}}}{ZB}}\ {\mbox{cm}}} \)

plasma skin depth (also called the electron inertial length), the depth in a plasma to which electromagnetic radiation can penetrate:
\( {\displaystyle {\frac {c}{\omega _{pe}}}\approx 5.31\times 10^{5}\,{\frac {1}{{n_{e}}^{\frac {1}{2}}}}\ {\mbox{cm}}} \)

Debye length, the scale over which electric fields are screened out by a redistribution of the electrons:
\( {\displaystyle \lambda _{D}=\left({\frac {kT_{e}}{4\pi ne^{2}}}\right)^{\frac {1}{2}}={\frac {v_{Te}}{\omega _{pe}}}\approx 7.43\times 10^{2}\,\left({\frac {T_{e}}{n}}\right)^{\frac {1}{2}}\ {\mbox{cm}}} \)

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:
\( {\displaystyle d_{i}={\frac {c}{\omega _{pi}}}\approx 2.28\times 10^{7}\,{\frac {1}{Z}}\left({\frac {\mu }{n_{i}}}\right)^{\frac {1}{2}}\ {\mbox{cm}}} \)

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

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

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

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

\( {\displaystyle v_{Ti}=\left({\frac {kT_{i}}{m_{i}}}\right)^{\frac {1}{2}}\approx 9.79\times 10^{5}\,\left({\frac {T_{i}}{\mu }}\right)^{\frac {1}{2}}\ {\mbox{cm/s}}} \)
ion speed of sound, the speed of the longitudinal waves resulting from the mass of the ions and the pressure of the electrons:

\( {\displaystyle c_{s}=\left({\frac {\gamma ZkT_{e}}{m_{i}}}\right)^{\frac {1}{2}}\approx 9.79\times 10^{5}\,\left({\frac {\gamma ZT_{e}}{\mu }}\right)^{\frac {1}{2}}\ {\mbox{cm/s}}} \) , 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:

\( {\displaystyle v_{A}={\frac {B}{\left(4\pi n_{i}m_{i}\right)^{\frac {1}{2}}}}\approx 2.18\times 10^{11}\,{\frac {B}{\left(\mu n_{i}\right)^{\frac {1}{2}}}}\ {\mbox{cm/s}}} \) in cgs units,
\( {\displaystyle v_{A}={\frac {B}{\left(\mu _{0}n_{i}m_{i}\right)^{\frac {1}{2}}}}} \) in SI units.

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

\( {\displaystyle \left({\frac {4\pi }{3}}\right)n\lambda _{D}^{3}\approx 1.72\times 10^{9}\,\left({\frac {T^{3}}{n}}\right)^{\frac {1}{2}}} \)

Alfvén speed to speed of light ratio

\( {\displaystyle {\frac {v_{A}}{c}}\approx 7.28\,{\frac {B}{\left(\mu n_{i}\right)^{\frac {1}{2}}}}} \)

electron plasma frequency to gyrofrequency ratio

\( {\displaystyle {\frac {\omega _{pe}}{\omega _{ce}}}\approx 3.21\times 10^{-3}\,{\frac {{n_{e}}^{\frac {1}{2}}}{B}}} \)

ion plasma frequency to gyrofrequency ratio

\( {\displaystyle {\frac {\omega _{pi}}{\omega _{ci}}}\approx 0.137\,{\frac {\left(\mu n_{i}\right)^{\frac {1}{2}}}{B}}} \)

thermal pressure to magnetic pressure ratio, or beta, β

\( {\displaystyle \beta ={\frac {8\pi nkT}{B^{2}}}\approx 4.03\times 10^{-11}\,{\frac {nT}{B^{2}}}} \)

magnetic field energy to ion rest energy ratio

\( {\displaystyle {\frac {B^{2}}{8\pi n_{i}m_{i}c^{2}}}\approx 26.5\,{\frac {B^{2}}{\mu n_{i}}}} \)

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 ν ∗ {\displaystyle \nu ^{*}} \nu ^{*} is defined as[3][4]

\( {\displaystyle \nu ^{*}=\nu _{\mathrm {ei} }\,{\sqrt {\frac {m_{\mathrm {e} }}{k_{\mathrm {B} }T_{\mathrm {e} }}}}\,{\frac {1}{\epsilon ^{\frac {3}{2}}}}\,qR,} \)

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.

Electron temperature

Temperature is a statistical quantity whose formal definition is

\( {\displaystyle T=\left({\frac {\partial U}{\partial S}}\right)_{V,N},} \)

or the change in internal energy with respect to entropy, holding volume and particle number constant. A practical definition comes from the fact that the atoms, molecules, or whatever particles in a system have an average kinetic energy. The average means to average over the kinetic energy of all the particles in a system.

If the velocities of a group of electrons, e.g., in a plasma, follow a Maxwell–Boltzmann distribution, then the electron temperature is defined as the temperature of that distribution. For other distributions, not assumed to be in equilibrium or have a temperature, two-thirds of the average energy is often referred to as the temperature, since for a Maxwell–Boltzmann distribution with three degrees of freedom, \( {\displaystyle \langle E\rangle =(3/2)\,k_{\text{B}}T} \).

The SI unit of temperature is the kelvin (K), but using the above relation the electron temperature is often expressed in terms of the energy unit electronvolt (eV). Each kelvin (1 K) corresponds to 8.617 333 262...×10−5 eV; this factor is the ratio of the Boltzmann constant to the elementary charge.[5] Each eV is equivalent to 11,605 kelvins, which can be calculated by the relation \( {\displaystyle \langle E\rangle =k_{\text{B}}T} \).

The electron temperature of a plasma can be several orders of magnitude higher than the temperature of the neutral species or of the ions. This is a result of two facts. Firstly, many plasma sources heat the electrons more strongly than the ions. Secondly, atoms and ions are much heavier than electrons, and energy transfer in a two-body collision is much more efficient if the masses are similar. Therefore, equilibration of the temperature happens very slowly, and is not achieved during the time range of the observation.