Plasma parameters: Difference between revisions

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{{Short description|Characteristic values of a plasma}}

{{Distinguish|text=the [[plasma parameter]]}}

[[Image:Magnetic rope.svg|thumb|300px|The complex self-constricting magnetic field lines and current paths in a [[Birkeland current]] that may develop in a plasma (''[https://history.nasa.gov/SP-345/ch15.htm#250 Evolution of the Solar System]'', 1976) ]]

'''Plasma parameters''' define various characteristics of a [[Plasma (physics)|plasma]], an electrically conductive collection of [[charged particle|charged]]s thatand responds ''collectively'' toneutral [[electromagnetic forceparticle]]s. Plasma typically takes the form of neutralvarious gas-likespecies clouds([[electron]]s or chargedand [[ion beam]]s,) butthat mayresponds also''collectively'' includeto dust[[electromagnetic and grainsforce]]s.<ref>Peratt, Anthony, ''Physics of the Plasma Universe'' (1992);</ref> The behaviour of suchSuch particle systems can be studied [[Statistical mechanics|statistically]], i.e., their behaviour can be described based on a limited number of global parameters instead of tracking each particle separately.<ref>Parks, George K., ''Physics of Space Plasmas'' (2004, 2nd Ed.)</ref>

== Fundamental plasma parameters ==

The fundamental plasma parameters in a [[steady state]] are

All quantities are in [[Gaussian units|Gaussian]] ([[Centimetre-gram-second system of units|cgs]]) units except [[energy]] and [[temperature]] expressed in eV and ion mass expressed in units of the [[proton]] mass <math>\mu = m_i/m_p</math>; <math>Z</math> is charge state; <math>k</math> is [[Boltzmann's constant]]; <math>K</math> is wavenumber; <math>\ln\Lambda</math> is the [[Coulomb logarithm]].

* the [[number density]] <math>n_s</math> of each particle species <math>s</math> present in the plasma,

* the [[temperature]] <math>T_s</math> of each species,

* the [[mass]] <math>m_s</math> of each species,

* the [[electric charge|charge]] <math>q_s</math> of each species,

* and the [[magnetic flux density]] <math>B</math>.

Using these parameters and [[physical constant]]s, other plasma parameters can be derived.<ref name="bellan06">{{cite book |last1=Bellan |first1=Paul Murray |title=Fundamentals of plasma physics |date=2006 |publisher=Cambridge University Press |___location=Cambridge |isbn=0521528003}}</ref>

== Other ==

All quantities are in [[Gaussian units|Gaussian]] ([[Centimetre-gram-second system of units|cgs]]) units except [[energy]] <math>E</math> and [[temperature]] expressed<math>T</math> which are in eV[[electronvolt]]s. andFor the sake of simplicity, a single ionic species is assumed. The ion mass is expressed in units of the [[proton]] mass, <math>\mu = m_i/m_p</math>; and the ion charge in units of the [[elementary charge]] {{nowrap|<math>e</math>,}} <math>Z = q_i/e</math> is(in chargethe state;case of a fully ionized atom, <math>kZ</math> isequals to the respective [[atomic number]]). The other physical quantities used are the [[Boltzmann's constant]]; {{nowrap|(<math>Kk_\text{B}</math>),}} is[[speed wavenumber;of light]] {{nowrap|(<math>\ln\Lambdac</math>),}} isand the [[Coulomb logarithm]] {{nowrap|(<math>\ln\Lambda</math>).}}

=== Frequencies ===

{{unordered list

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

: <math display="block">\omega_{ce} = \frac{eB}{m_e c} \approx 1.76 \times 10^{7}\,B\ \mbox{rad/s} \,</math>

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

: <math display="block">\omega_{ci} = \frac{ZeB}{m_i c} \approx 9.58 \times 10^3\,\frac{ZB}{\mu}\ \mbox{rad/s} \,</math>

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

: <math display="block">\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}</math>

| '''ion plasma frequency''':

: <math display="block">\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}</math>

| '''electron trapping rate''':

: <math display="block">\nu_{Te} = \left(\frac{eKEe E}{m_e}\right)^\frac{1}{2} \approx 7.26 \times 10^8\,\left(KE\right)E^\frac{1}{2}\ /\mboxmathrm{s^{-1}} \,</math>

| '''ion trapping rate''':

: <math display="block">\nu_{Ti} = \left(\frac{ZeKEZ e E}{m_i}\right)^\frac{1}{2} \approx {1.69 \times 10^7}\,\left(\frac{ZKEZ E}{\mu}\right)^\frac{1}{2}\ /\mboxmathrm{s^{-1}} \,</math>

| '''electron collision rate in completely ionized plasmas''':

: <math display="block">\nu_e \approx 2.91 \times 10^{-6}\,\frac{n_e\ln\Lambda}{T_e^\frac{3}{2}}\ /\mboxmathrm{s^{-1}}</math>

| '''ion collision rate in completely ionized plasmas''':

: <math display="block">\nu_i \approx 4.80 \times 10^{-8}\,\frac{Z^4 n_i\,\ln\Lambda}{\left(T_i^3 \mu\right)^\frac{1}{2}} \ /\mboxmathrm{s^{-1}}</math>

}}

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{{unordered list

| '''[[Thermal de Broglie wavelength|electron thermal de Broglie wavelength]]''', approximate average [[de Broglie wavelength]] of electrons in a plasma:

: <math display="block">\lambda_{\mathrm{th},e} = \sqrt{\frac{h^2}{2\pi m_e kT_ek_\text{B} T_e}} \approx 6.919 \times 10^{-8}\,\frac{1}{{T_e}^\frac{1}{2}}\ \mbox{cm}</math>

| '''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:

: <math display="block">\frac{e^2}{kTk_\text{B} T} \approx 1.44 \times 10^{-7}\,\frac{1}{T}\ \mbox{cm}</math>

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

: <math display="block">r_e = \frac{v_{Te}}{\omega_{ce}} \approx 2.38\,\frac{{T_e}^\frac{1}{2}}{B}\ \mbox{cm}</math>

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

: <math display="block">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}</math>

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

: <math display="block">\frac{c}{\omega_{pe}} \approx 5.31 \times 10^5\,\frac{1}{{n_e}^\frac{1}{2}}\ \mbox{cm}</math>

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

: <math display="block">\lambda_D = \left(\frac{kT_ek_\text{B} T_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}</math>

| '''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:

: <math display="block">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}</math>

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

: <math display="block">\lambda_{e,i} = \frac{\overline{v_{e,i}}}{\nu_{e,i}},</math>,

where <math>\overline{v_{e,i}}</math> is an average velocity of the electron (ion) and <math>\nu_{e,i}</math> is the electron or ion '''collision rate'''.

}}

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{{unordered list

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

: <math display="block">v_\text{Teth,e} = \left(\frac{kT_ek_\text{B} T_e}{m_e}\right)^\frac{1}{2} \approx 4.19 \times 10^7\,{T_e}^\frac{1}{2} \ \mbox{cm/s}</math>

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

: <math display="block">v_\text{Tith,i} = \left(\frac{kT_ik_\text{B} T_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}</math>

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

: <math display="block">c_s = \left(\frac{\gamma ZkT_eZ k_\text{B} T_e}{m_i}\right)^\frac{1}{2} \approx 9.79 \times 10^5\,\left(\frac{\gamma ZT_eZ T_e}{\mu}\right)^\frac{1}{2}\ \mbox{cm/s},</math>,

where <math>\gamma</math> is the [[adiabatic index]]

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=== Dimensionless ===

* number of particles in a Debye sphere <math display="block">\frac{4\pi}{3} n\lambda_\text{D}^3 \approx 1.72 \times 10^9 \, \left(\frac{T^3}{n}\right)^\frac{1}{2}</math>

[[Image:fusor running.jpg|thumb|right|300px|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.]]

*: Alfvén speed to speed of light ratio <math display="block">\frac{\omega_{pi}v_A}{\omega_{ci}c} \approx 07.13728\,\frac{B}{\left(\mu n_i\right)^\frac{1}{2}}{B}</math>

* number of particles in a Debye sphere

*: electron plasma frequency to gyrofrequency ratio <math display="block">\left(\frac{4\pi}omega_{3pe}}{\right)n\lambda_D^3omega_{ce}} \approx 13.7221 \times 10^9{-3}\,\left(\frac{T^3}{nn_e}\right)^\frac{1}{2}}{B}</math>

*: ion plasma frequency to gyrofrequency ratio <math display="block">\frac{\omega_{pepi}}{\omega_{ceci}} \approx 30.21 \times 10^{-3}137\,\frac{{n_e}\left(\mu n_i\right)^\frac{1}{2}}{B}</math>

* Alfvén speed to speed of light ratio

*: thermal pressure to magnetic pressure ratio, or [[beta (plasma physics)|beta]], ''β'' <math display="block">\beta = \frac{v_A8\pi n k_\text{B} T}{cB^2} \approx 74.2803 \,\fractimes 10^{B-11}{\left(\mu n_i\right)^,\frac{1nT}{B^2}}</math>

* [[magnetic energy|magnetic field energy]] to [[invariant mass#Rest energy|ion rest energy]] ratio <math display="block">\frac{B^2}{8\pi n_i m_i c^2} \approx 26.5\,\frac{B^2}{\mu n_i}</math>

* electron plasma frequency to gyrofrequency ratio

*: <math>\frac{\omega_{pe}}{\omega_{ce}} \approx 3.21 \times 10^{-3}\,\frac{{n_e}^\frac{1}{2}}{B}</math>

* ion plasma frequency to gyrofrequency ratio

*: <math>\frac{\omega_{pi}}{\omega_{ci}} \approx 0.137\,\frac{\left(\mu n_i\right)^\frac{1}{2}}{B}</math>

* thermal pressure to magnetic pressure ratio, or [[beta (plasma physics)|beta]], β

*: <math>\beta = \frac{8\pi nkT}{B^2} \approx 4.03 \times 10^{-11}\,\frac{nT}{B^2}</math>

* [[magnetic energy|magnetic field energy]] to [[invariant mass#Rest energy|ion rest energy]] ratio

*: <math>\frac{B^2}{8\pi n_i m_i c^2} \approx 26.5\,\frac{B^2}{\mu n_i}</math>

==Collisionality==

In the study of [[tokamak]]s, '''collisionality''' is a [[dimensionless parameter]] which expresses the ratio of the electron-ion [[collision frequency]] to the [[banana orbit]] frequency.

The [[Plasma (physics)|plasma]] collisionality <math>\nu^*</math> is defined as<ref>{{ 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 }}</ref><ref>{{ 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 | url-access = subscription }}</ref>

The [[Plasma (physics)|plasma]] collisionality <math>\nu^*</math> is defined as<ref>Nucl. Fusion, Vol. 39, No. 12 (1999)</ref><ref>Wenzel, K and Sigmar, D.. Nucl. Fusion 30, 1117 (1990)</ref>

<math display="block">

\nu^* = \nu_\mathrm{ei} \, \sqrt{\frac{m_\mathrm{e}}{k_\mathrm{B} T_\mathrm{e}}} \, \frac{1}varepsilon^{\epsilon^-\frac{3}{2}} \, qR,

:<math>

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

</math>

where <math>\nu_\mathrm{ei}</math> denotes the electron-ion [[collision frequency]], <math>R</math> is the major radius of the plasma, <math>\epsilonvarepsilon</math> is the inverse [[aspect-ratio]], and <math>q</math> is the [[safety factor]]. The [[Plasma (physics)|plasma]] parameters <math>m_\mathrm{i}</math> and <math>T_\mathrm{i}</math> denote, respectively, the [[mass]] and [[temperature]] of the [[ions]], and <math>k_\mathrm{B}</math> is the [[Boltzmann constant]].

where <math>\nu_\mathrm{ei}</math> denotes the electron-ion [[collision frequency]], <math>R</math> is the major radius of the plasma, <math>\epsilon</math> is the inverse [[aspect-ratio]], and <math>q</math> is the [[safety factor]]. The [[Plasma (physics)|plasma]] parameters <math>m_\mathrm{i}</math> and <math>T_\mathrm{i}</math> denote, respectively, the [[mass]] and [[temperature]] of the [[ions]], and <math>k_\mathrm{B}</math> is the [[Boltzmann constant]].

==Electron temperature==

Temperature is a statistical quantity whose formal definition is

:<math display="block">T = \left(\frac{\partial U}{\partial S}\right)_{V,N},</math>

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 [[velocity|velocities]] of a group of [[electron]]s, e.g., in a [[plasma (physics)|plasma]], follow a [[Maxwell–Boltzmann distribution#Distribution of the velocity vector|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 (physics and chemistry)|degrees of freedom]], <math display="inline">\langle E \rangle = (\frac 3/ 2) \, k_\text{B} T</math>.

The [[International System of Units|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&nbsp;K) corresponds to {{val|8.617&nbsp;333&nbsp;262617333262|end=...×10⁻⁵&nbsp;|e=-5|u=eV}}; this factor is the ratio of the [[Boltzmann constant]] to the [[elementary charge]].<ref name=NIST>{{ 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 }}

{{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 <i>x</i> for relating K to eV

|last=Mohr

|first=Peter J.

|last2=Newell

|first2=David B.

|last3=Taylor

|first3=Barry N.

|last4=Tiesenga

|first4=E.

|date=20 May 2019

|website=The NIST Reference on Constants, Units, and Uncertainty

|publisher=National Institute of Standards and Technology

|access-date=11 November 2019

|quote= }}

</ref> Each eV is equivalent to 11,605 [[kelvin]]s, which can be calculated by the relation <math>\langle E \rangle = k_\text{B} T</math>.

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==See also==

*[[List of plasma physics articles]]

* [[Ball-pen probe]]

* [[Langmuir probe]]

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==References==

{{reflist}}

*[https://www.nrl.navy.mil/ppdNews-Media/sitesPublications/www.nrl.navy.mil.ppdNRL-Plasma-Formulary/files/pdfs/NRL_FORMULARY_18.pdf NRL Plasma Formulary] – [[United States Naval Research Laboratory|Naval Research Laboratory]] (2018)

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