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Undid revision 1269574160 by Vbrcat (talk) It's clearly said the temperature is in eV |
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== Other ==
All quantities are in [[Gaussian units|Gaussian]] ([[Centimetre-gram-second system of units|cgs]]) units except [[energy]] <math>E</math> and [[temperature]] <math>T</math> which are in [[electronvolt]]s. For 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]] <math>e</math>, <math>Z = q_i/e</math> (in the case of a fully ionized atom, <math>Z</math> equals to the respective [[atomic number]]). The other physical quantities used are the [[Boltzmann constant]] (<math>k</math>), [[speed of light]] (<math>c</math>), and the [[Coulomb logarithm]] (<math>\ln\Lambda</math>).
=== Frequencies ===
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<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{
| '''ion trapping rate''':
<math display="block">\nu_{Ti} = \left(\frac{
| '''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>
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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
| '''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>
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<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>
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=== Dimensionless ===
* 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">\frac{v_A}{c} \approx 7.28\,\frac{B}{\left(\mu n_i\right)^\frac{1}{2}}</math>
* electron plasma frequency to gyrofrequency ratio <math display="block">\frac{\omega_{pe}}{\omega_{ce}} \approx 3.21 \times 10^{-3}\,\frac{{n_e}^\frac{1}{2}}{B}</math>
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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 Editors | 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 }}</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 }}</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>
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]].
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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 K) corresponds to {{val|8.617333262|end=...|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 }}
</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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