Plasma parameters#Fundamental plasma parameters

The fundamental plasma parameters in a steady state are

• the number density $n\_s$ of each particle species $s$ present in the plasma,
• the temperature $T\_s$ of each species,
• the mass $m\_s$ of each species,
• the charge $q\_s$ of each species,
• and the magnetic flux density $B$.

Using these parameters and physical constants, other plasma parameters can be derived.{{cite book |last1=Bellan |first1=Paul Murray |title=Fundamentals of plasma physics |date=2006 |publisher=Cambridge University Press |location=Cambridge |isbn=0521528003}}

All quantities are in Gaussian (cgs) units except energy $E$ and temperature $T$ which are in electronvolts. For the sake of simplicity, a single ionic species is assumed. The ion mass is expressed in units of the proton mass, $\backslash mu\; =\; m\_i/m\_p$ and the ion charge in units of the elementary charge {{nowrap|$e$,}} $Z\; =\; q\_i/e$ (in the case of a fully ionized atom, $Z$ equals to the respective atomic number). The other physical quantities used are the Boltzmann constant {{nowrap|($k\_\backslash text\{B\}$),}} speed of light {{nowrap|($c$),}} and the Coulomb logarithm {{nowrap|($\backslash ln\backslash Lambda$).}}

{{unordered list

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

$$\backslash omega\_\{ce\}\; =\; \backslash frac\{eB\}\{m\_e\; c\}\; \backslash approx\; 1.76\; \backslash times\; 10^\{7\}\backslash ,B\backslash \; \backslash mbox\{rad/s\}$$

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

$$\backslash omega\_\{ci\}\; =\; \backslash frac\{ZeB\}\{m\_i\; c\}\; \backslash approx\; 9.58\; \backslash times\; 10^3\backslash ,\backslash frac\{ZB\}\{\backslash mu\}\backslash \; \backslash mbox\{rad/s\}$$

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

$$\backslash omega\_\{pe\}\; =\; \backslash left(\backslash frac\{4\; \backslash pi\; n\_e\; e^2\}\{m\_e\}\backslash right)^\backslash frac\{1\}\{2\}\; \backslash approx\; 5.64\; \backslash times\; 10^4\backslash ,\{n\_e\}^\backslash frac\{1\}\{2\}\; \backslash \; \backslash mbox\{rad/s\}$$

| ion plasma frequency:

$$\backslash omega\_\{pi\}\; =\; \backslash left(\backslash frac\{4\backslash pi\; n\_i\; Z^2\; e^2\}\{m\_i\}\backslash right)^\backslash frac\{1\}\{2\}\; \backslash approx\; \{1.32\; \backslash times\; 10^3\}\; \backslash ,Z\; \backslash left(\backslash frac\{n\_i\}\{\backslash mu\}\backslash right)^\backslash frac\{1\}\{2\}\backslash \; \backslash mbox\{rad/s\}$$

| electron trapping rate:

$$\backslash nu\_\{Te\}\; =\; \backslash left(\backslash frac\{e\; E\}\{m\_e\}\backslash right)^\backslash frac\{1\}\{2\}\; \backslash approx\; 7.26\; \backslash times\; 10^8\backslash ,E^\backslash frac\{1\}\{2\}\backslash \; \backslash mathrm\{s^\{-1\}\}$$

| ion trapping rate:

$$\backslash nu\_\{Ti\}\; =\; \backslash left(\backslash frac\{Z\; e\; E\}\{m\_i\}\backslash right)^\backslash frac\{1\}\{2\}\; \backslash approx\; \{1.69\; \backslash times\; 10^7\}\backslash ,\backslash left(\backslash frac\{Z\; E\}\{\backslash mu\}\backslash right)^\backslash frac\{1\}\{2\}\backslash \; \backslash mathrm\{s^\{-1\}\}$$

| electron collision rate in completely ionized plasmas:

$$\backslash nu\_e\; \backslash approx\; 2.91\; \backslash times\; 10^\{-6\}\backslash ,\backslash frac\{n\_e\backslash ln\backslash Lambda\}\{T\_e^\backslash frac\{3\}\{2\}\}\backslash \; \backslash mathrm\{s^\{-1\}\}$$

| ion collision rate in completely ionized plasmas:

$$\backslash nu\_i\; \backslash approx\; 4.80\; \backslash times\; 10^\{-8\}\backslash ,\backslash frac\{Z^4\; n\_i\backslash ,\backslash ln\backslash Lambda\}\{\backslash left(T\_i^3\; \backslash mu\backslash right)^\backslash frac\{1\}\{2\}\}\; \backslash \; \backslash mathrm\{s^\{-1\}\}$$

}}

{{unordered list

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

$$\backslash lambda\_\{\backslash mathrm\{th\},e\}\; =\; \backslash sqrt\{\backslash frac\{h^2\}\{2\backslash pi\; m\_e\; k\_\backslash text\{B\}\; T\_e\}\}\; \backslash approx\; 6.919\; \backslash times\; 10^\{-8\}\backslash ,\backslash frac\{1\}\{\{T\_e\}^\backslash frac\{1\}\{2\}\}\backslash \; \backslash mbox\{cm\}$$

| classical distance of closest approach, also known as "Landau length" 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:

$$\backslash frac\{e^2\}\{k\_\backslash text\{B\}\; T\}\; \backslash approx\; 1.44\; \backslash times\; 10^\{-7\}\backslash ,\backslash frac\{1\}\{T\}\backslash \; \backslash mbox\{cm\}$$

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

$$r\_e\; =\; \backslash frac\{v\_\{Te\}\}\{\backslash omega\_\{ce\}\}\; \backslash approx\; 2.38\backslash ,\backslash frac\{\{T\_e\}^\backslash frac\{1\}\{2\}\}\{B\}\backslash \; \backslash mbox\{cm\}$$

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

$$r\_i\; =\; \backslash frac\{v\_\{Ti\}\}\{\backslash omega\_\{ci\}\}\; \backslash approx\; 1.02\; \backslash times\; 10^2\backslash ,\backslash frac\{\backslash left(\backslash mu\; T\_i\backslash right)^\backslash frac\{1\}\{2\}\}\{ZB\}\backslash \; \backslash mbox\{cm\}$$

| plasma skin depth (also called the electron inertial length), the depth in a plasma to which electromagnetic radiation can penetrate:

$$\backslash frac\{c\}\{\backslash omega\_\{pe\}\}\; \backslash approx\; 5.31\; \backslash times\; 10^5\backslash ,\backslash frac\{1\}\{\{n\_e\}^\backslash frac\{1\}\{2\}\}\backslash \; \backslash mbox\{cm\}$$

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

$$\backslash lambda\_D\; =\; \backslash left(\backslash frac\{k\_\backslash text\{B\}\; T\_e\}\{4\backslash pi\; ne^2\}\backslash right)^\backslash frac\{1\}\{2\}\; =\; \backslash frac\{v\_\{Te\}\}\{\backslash omega\_\{pe\}\}\; \backslash approx\; 7.43\; \backslash times\; 10^2\backslash ,\backslash left(\backslash frac\{T\_e\}\{n\}\backslash right)^\backslash frac\{1\}\{2\}\backslash \; \backslash mbox\{cm\}$$

| 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:

$$d\_i\; =\; \backslash frac\{c\}\{\backslash omega\_\{pi\}\}\; \backslash approx\; 2.28\; \backslash times\; 10^7\backslash ,\; \backslash frac\{1\}\{Z\}\; \backslash left(\backslash frac\{\backslash mu\}\{n\_i\}\backslash right)^\backslash frac\{1\}\{2\}\backslash \; \backslash mbox\{cm\}$$

| mean free path, the average distance between two subsequent collisions of the electron (ion) with plasma components:

$$\backslash lambda\_\{e,i\}\; =\; \backslash frac\{\backslash overline\{v\_\{e,i\}\}\}\{\backslash nu\_\{e,i\}\},$$

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

}}

{{unordered list

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

$$v\_\backslash text\{th,e\}\; =\; \backslash left(\backslash frac\{k\_\backslash text\{B\}\; T\_e\}\{m\_e\}\backslash right)^\backslash frac\{1\}\{2\}\; \backslash approx\; 4.19\; \backslash times\; 10^7\backslash ,\{T\_e\}^\backslash frac\{1\}\{2\}\; \backslash \; \backslash mbox\{cm/s\}$$

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

$$v\_\backslash text\{th,i\}\; =\; \backslash left(\backslash frac\{k\_\backslash text\{B\}\; T\_i\}\{m\_i\}\backslash right)^\backslash frac\{1\}\{2\}\; \backslash approx\; 9.79\; \backslash times\; 10^5\backslash ,\backslash left(\backslash frac\{T\_i\}\{\backslash mu\}\backslash right)^\backslash frac\{1\}\{2\}\backslash \; \backslash mbox\{cm/s\}$$

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

$$c\_s\; =\; \backslash left(\backslash frac\{\backslash gamma\; Z\; k\_\backslash text\{B\}\; T\_e\}\{m\_i\}\backslash right)^\backslash frac\{1\}\{2\}\; \backslash approx\; 9.79\; \backslash times\; 10^5\backslash ,\backslash left(\backslash frac\{\backslash gamma\; Z\; T\_e\}\{\backslash mu\}\backslash right)^\backslash frac\{1\}\{2\}\backslash \; \backslash mbox\{cm/s\},$$

where $\backslash gamma$ 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:

: $v\_A\; =\; \backslash frac\{B\}\{\backslash left(4\backslash pi\; n\_i\; m\_i\backslash right)^\backslash frac\{1\}\{2\}\}\; \backslash approx\; 2.18\; \backslash times\; 10^\{11\}\backslash ,\backslash frac\{B\}\{\backslash left(\backslash mu\; n\_i\backslash right)^\backslash frac\{1\}\{2\}\}\backslash \; \backslash mbox\{cm/s\}$ in cgs units,

: $v\_A\; =\; \backslash frac\{B\}\{\backslash left(\backslash mu\_0\; n\_i\; m\_i\backslash right)^\backslash frac\{1\}\{2\}\}$ in SI units.

}}

• number of particles in a Debye sphere $$\backslash frac\{4\backslash pi\}\{3\}\; n\backslash lambda\_\backslash text\{D\}^3\; \backslash approx\; 1.72\; \backslash times\; 10^9\; \backslash ,\; \backslash left(\backslash frac\{T^3\}\{n\}\backslash right)^\backslash frac\{1\}\{2\}$$
• Alfvén speed to speed of light ratio $$\backslash frac\{v\_A\}\{c\}\; \backslash approx\; 7.28\backslash ,\backslash frac\{B\}\{\backslash left(\backslash mu\; n\_i\backslash right)^\backslash frac\{1\}\{2\}\}$$
• electron plasma frequency to gyrofrequency ratio $$\backslash frac\{\backslash omega\_\{pe\}\}\{\backslash omega\_\{ce\}\}\; \backslash approx\; 3.21\; \backslash times\; 10^\{-3\}\backslash ,\backslash frac\{\{n\_e\}^\backslash frac\{1\}\{2\}\}\{B\}$$
• ion plasma frequency to gyrofrequency ratio $$\backslash frac\{\backslash omega\_\{pi\}\}\{\backslash omega\_\{ci\}\}\; \backslash approx\; 0.137\backslash ,\backslash frac\{\backslash left(\backslash mu\; n\_i\backslash right)^\backslash frac\{1\}\{2\}\}\{B\}$$
• thermal pressure to magnetic pressure ratio, or beta, β $$\backslash beta\; =\; \backslash frac\{8\backslash pi\; n\; k\_\backslash text\{B\}\; T\}\{B^2\}\; \backslash approx\; 4.03\; \backslash times\; 10^\{-11\}\backslash ,\backslash frac\{nT\}\{B^2\}$$
• magnetic field energy to ion rest energy ratio $$\backslash frac\{B^2\}\{8\backslash pi\; n\_i\; m\_i\; c^2\}\; \backslash approx\; 26.5\backslash ,\backslash frac\{B^2\}\{\backslash mu\; n\_i\}$$

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 $\backslash nu^*$ is defined as{{ cite journal | author1 = ITER Physics Expert Group on Diagnostics | author2 = ITER Physics Basis | date = 1999 | title = Chapter 7: Measurement of plasma parameters | url = https://iopscience.iop.org/article/10.1088/0029-5515/39/12/307 | journal = Nuclear Fusion | volume = 39 | issue = 12 | pages = 2541–2575 | doi = 10.1088/0029-5515/39/12/307 | issn = 0029-5515 | url-access = subscription }}{{ cite journal | last1 = Wenzel | first1 = K.W. | last2 = Sigmar | first2 = D.J. | date = 1990-06-01 | title = Neoclassical analysis of impurity transport following transition to improved particle confinement | url = https://iopscience.iop.org/article/10.1088/0029-5515/30/6/013 | journal = Nuclear Fusion | volume = 30 | issue = 6 |pages = 1117–1127 | doi = 10.1088/0029-5515/30/6/013 | issn = 0029-5515 | hdl = 1721.1/95018 | hdl-access = free }}

\nu^* = \nu_\mathrm{ei} \, \sqrt{\frac{m_\mathrm{e}}{k_\mathrm{B} T_\mathrm{e}}} \, \varepsilon^{-\frac{3}{2}} \, qR,

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

Temperature is a statistical quantity whose formal definition is

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

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, $\backslash langle\; E\; \backslash rangle\; =\; \backslash frac\; 3\; 2\; \backslash ,\; k\_\backslash text\{B\}\; T$.

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 {{val|8.617333262|end=...|e=-5|u=eV}}; this factor is the ratio of the Boltzmann constant to the elementary charge.{{ cite web | url = https://physics.nist.gov/cgi-bin/cuu/Convert?exp=0&num=1&From=k&To=ev&Action=Only+show+factor | title = CODATA Energy conversion factor: Factor x for relating K to eV | last1 = Mohr | first1 = Peter J. | last2 = Newell | first2 = David B. | last3 = Taylor | first3 = Barry N. | last4 = Tiesenga | first4 = E. | date = 2019-05-20 | website = The NIST Reference on Constants, Units, and Uncertainty | publisher = National Institute of Standards and Technology | access-date = 2019-11-11 }}

Each eV is equivalent to 11,605 kelvins, which can be calculated by the relation $\backslash langle\; E\; \backslash rangle\; =\; k\_\backslash text\{B\}\; T$.

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.

• Ball-pen probe
• Langmuir probe

{{reflist}}

• [https://www.nrl.navy.mil/News-Media/Publications/NRL-Plasma-Formulary/ NRL Plasma Formulary] – Naval Research Laboratory (2018)

Plasma parameters

Category:Astrophysics