Plasma parameters: Difference between revisions

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{{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{eKE}{m_e}\right)^\frac{1}{2} \approx 7.26 \times 10^8\,\left(KE\right)^\frac{1}{2}\ /\mbox{s} \,</math>
| '''ion trapping rate''':
: <math display="block">\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} \,</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}}\ /\mbox{s}</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}} \ /\mbox{s}</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_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}{kT} \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_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_{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}</math>
| '''ion thermal velocity''', typical velocity of an ion in a [[Maxwell–Boltzmann distribution]]:
: <math display="block">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}</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_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},</math>,
 
where <math>\gamma</math> is the [[adiabatic index]]
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=== Dimensionless ===
[[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.]]
* number of particles in a Debye sphere <math display="block">\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}</math>
*: Alfvén speed to speed of light ratio <math display="block">\left(\frac{4\piv_A}{3c}\right)n\lambda_D^3 \approx 17.72 \times 10^928\,\left(\frac{T^3B}{n}\left(\mu n_i\right)^\frac{1}{2}}</math>
*: electron plasma frequency to gyrofrequency ratio <math display="block">\frac{\omega_{pipe}}{\omega_{cice}} \approx 03.13721 \times 10^{-3}\,\frac{\left(\mu n_i\right){n_e}^\frac{1}{2}}{B}</math>
* Alfvén speed to speed of light ratio
*: ion plasma frequency to gyrofrequency ratio <math display="block">\frac{v_A\omega_{pi}}{c\omega_{ci}} \approx 70.28137\,\frac{B}{\left(\mu n_i\right)^\frac{1}{2}}{B}</math>
*: thermal pressure to magnetic pressure ratio, or [[beta (plasma physics)|beta]], β <math display="block">\beta = \frac{8\pi nkT}{B^2} \approx 4.03 \times 10^{-11}\,\frac{nT}{B^2}</math>
* electron plasma frequency to gyrofrequency ratio
*: [[magnetic energy|magnetic field energy]] to [[invariant mass#Rest energy|ion rest energy]] ratio <math display="block">\frac{\omega_{pe}B^2}{8\omega_{ce}pi n_i m_i c^2} \approx 326.21 \times 10^{-3}5\,\frac{{n_e}B^\frac{1}{2}}{B\mu n_i}</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==
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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}{\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>\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<sup>−5</sup>&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