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en>JRSpriggs: Undid revision 617100043 by James Waddington (talk) generally we do assume ZFC, so this is just a distraction here

2014-07-16T00:36:00Z

Undid revision 617100043 by James Waddington (talk) generally we do assume ZFC, so this is just a distraction here

← Older revision		Revision as of 02:36, 16 July 2014
Line 1:		Line 1:
	[[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 (''[http://history.nasa.gov/SP-345/ch15.htm#250 Evolution of the Solar System]'', 1976) ]]		Hello, my name is Andrew and my spouse doesn't like it at all. Distributing production has been his profession for some time. As a lady what she really likes is style and she's been performing it for fairly a while. My wife and I reside in Kentucky.<br><br>Here is my blog psychic readers ([http://www.youronlinepublishers.com/authWiki/AdolphvhBladenqq http://www.youronlinepublishers.com/])

	'''Plasma parameters''' define various characteristics of a [[Plasma (physics)\|plasma]], an electrically conductive collection of [[charged particle]]s that responds ''collectively'' to [[electromagnetic force]]s. Plasma typically takes the form of neutral gas-like clouds or charged [[ion beam]]s, but may also include dust and grains. <ref>Peratt, Anthony, ''Physics of the Plasma Universe'' (1992); </ref> The behaviour of such particle systems can be studied statistically. <ref>Parks, ~~George K., Physics of Space Plasmas (2004, 2nd Ed.)</ref>~~

	~~== Fundamental plasma parameters ==~~

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

	~~=== Frequencies ===~~
	*'''electron gyrofrequency''', the angular frequency of the circular motion of an electron in the plane perpendicular to the magnetic field:
	~~:<math>\omega_{ce} = eB/m_ec = 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>\omega_{ci} = eB/m_ic = 9.58 \times 10^3 Z \mu^{-1} B \mbox{rad/s} \,</math>~~
	*'''electron plasma frequency''', the frequency with which electrons oscillate ([[plasma oscillation]]):
	~~:<math>\omega_{pe} = (4\pi n_ee^2/m_e)^{1/2} = 5.64 \times 10^4 n_e^{1/2} \mbox{rad/s}</math>~~
	*'''ion plasma frequency''':
	~~:<math>\omega_{pi} = (4\pi n_iZ^2e^2/m_i)^{1/2} = 1.32 \times 10^3 Z \mu^{-1/2} n_i^{1/2} \mbox{rad/s}</math>~~
	*'''electron trapping rate''':
	~~:<math>\nu_{Te} = (eKE/m_e)^{1/2} = 7.26 \times 10^8 K^{1/2} E^{1/2} \mbox{s}^{-1} \,</math>~~
	*'''ion trapping rate''':
	~~:<math>\nu_{Ti} = (ZeKE/m_i)^{1/2} = 1.69 \times 10^7 Z^{1/2} K^{1/2} E^{1/2} \mu^{-1/2} \mbox{s}^{-1} \,</math>~~
	*'''electron collision rate''':
	~~:<math>\nu_e = 2.91 \times 10^{-6} n_e\,\ln\Lambda\,T_e^{-3/2} \mbox{s}^{-1}</math>~~
	*'''ion collision rate''':
	~~:<math>\nu_i = 4.80 \times 10^{-8} Z^4 \mu^{-1/2} n_i\,\ln\Lambda\,T_i^{-3/2} \mbox{s}^{-1}</math>~~

	~~=== Lengths ===~~
	*'''[[Thermal de Broglie wavelength\|Electron thermal de Broglie wavelength]]''', approximate average [[de Broglie wavelength]] of electrons in a plasma:
	~~:<math>\Lambda_e= \sqrt{\frac{h^2}{2\pi m_ekT_e}}= 6.919\times 10^{-8}\,T_e^{-1/2}\,\mbox{cm}</math>~~
	*'''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 have a velocity typical of the temperature, ignoring quantum-mechanical effects:~~
	~~:<math>e^2/kT=1.44\times10^{-7}\,T^{-1}\,\mbox{cm}</math>~~
	*'~~''electron gyroradius''', the radius of the circular motion of an electron in the plane perpendicular to the magnetic field:~~
	~~:<math>r_e = v_{Te}/\omega_{ce} = 2~~.~~38\,T_e^{1/2}B^{-1}\,\mbox{cm}</math>~~
	*'''ion gyroradius''', the radius of the circular motion of an ion in the plane perpendicular to the magnetic field:
	~~:<math>r_i = v_{Ti}/\omega_{ci} = 1~~.~~02\times10^2\,\mu^{1/2}Z^{-1}T_i^{1/2}B^{-1}\,\mbox{cm}</math>~~
	*'''plasma [[skin depth]]''', the depth in a ~~plasma to which electromagnetic radiation can penetrate:~~
	~~:<math>c/\omega_{pe} = 5.31\times10^5\,n_e^{-1/2}\,\mbox{cm}</math>~~
	*'''[[Debye length]]''', the scale over which electric fields are screened out by a redistribution of the electrons:
	~~:<math>\lambda_D = (kT/4\pi ne^2)^{1/2} = 7.43\times10^2\,T^{1/2}n^{-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>d_i = c/\omega_{pi}</math>~~

	~~=== Velocities ===~~
	*'''electron thermal velocity'''~~, typical velocity of an electron in a [[Maxwell-Boltzmann distribution]]:~~
	~~:<math>v_{Te} = (kT_e/m_e)^{1/2} = 4.19\times10^7\,T_e^{1/2}\,\mbox{cm/~~s~~}</math>~~
	*'''ion thermal velocity''', typical velocity of an ion in a ~~[[Maxwell-Boltzmann distribution]]:~~
	~~:<math>v_{Ti} = (kT_i/m_i)^{1/2} = 9.79\times10^5\,\mu^{-1/2}T_i^{1/2}\,\mbox{cm/s}</math>~~
	*'''ion sound velocity''', the speed of the longitudinal waves resulting from the mass of the ions and the pressure of the electrons:
	~~:<math>c_s = (\gamma ZkT_e/m_i)^{1/2} = 9~~.~~79\times10^5\,(\gamma ZT_e/\mu)^{1/2}\,\mbox{cm/s}</math>~~
	*'''[[Hannes Alfvén\|Alfvén]] velocity''', the speed of the [[Alfvén wave\|waves]] resulting from the mass of the ions and ~~the restoring force of the magnetic field:~~
	~~:<math>v_A = B/(4\pi n_im_i)^{1/2} = 2.18\times10^{11}\,\mu^{-1/2}n_i^{-1/2}B\,\mbox{cm/s}</math>~~

	~~=== 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~~.]]
	*square root of electron/proton mass ratio
	:<~~math~~>~~(m_e/m_p)^{1/2} = 2.33\times10^{-2} = 1/42.9 \,~~<~~/math~~>
	* number of particles in a Debye sphere
	~~:<math>(4\pi/3)n\lambda_D^3 = 1.72\times10^9\,T^{3/2}n^{-1/2}</math>~~
	* Alfvén velocity/speed of light
	~~:<math>v_A/c = 7.28\,\mu^{-1/2}n_i^{-1/2}B</math>~~
	* electron plasma/gyrofrequency ratio
	~~:<math>\omega_{pe}/\omega_{ce} = 3.21\times10^{-3}\,n_e^{1/2}B^{-1}</math>~~
	* ion plasma/gyrofrequency ratio
	~~:<math>\omega_{pi}/\omega_{ci} = 0.137\,\mu^{1/2}n_i^{1/2}B^{-1}</math>~~
	* thermal/magnetic pressure ratio (~~"[[Beta (plasma physics)\|beta]]")~~
	~~:<math>\beta = 8\pi nkT/B^2 = 4.03\times10^{-11}\,nTB^{-2}</math>~~
	* magnetic/ion rest energy ratio
	~~:<math>B^2/8\pi n_im_ic^2 = 26.5\,\mu^{-1}n_i^{-1}B^2</math>~~

	~~==See also==~~
	*[[List of plasma (physics) articles]]

	~~==References==~~
	* [http://www.ipp.mpg.de/~dpc/nrl/ NRL Plasma Formulary] (esp. [http://www.~~ipp~~.~~mpg.de~~/~~~dpc~~/~~nrl/28.html p. 28] and [~~http://www.~~ipp.mpg~~.~~de/~dpc/nrl~~/~~29.html p. 29~~])~~, J.D. Huba, [[United States Naval Research Laboratory\|Naval Research Laboratory]] (2007)~~

	~~==Footnotes==~~
	~~<div style="font-size: 95%">~~
	~~<!--See [[Wikipedia:Footnotes]] for an explanation of how to generate footnotes using the <ref(erences/)> tags-->~~
	~~<references/>~~
	~~</div>~~

	~~<!--Categories-->~~
	~~[[Category:Plasma physics\| ]]~~
	~~[[Category:Plasma diagnostics]]~~
	~~[[Category:Astrophysics]]~~

en>JRSpriggs: /* Rational numbers */ fix awkward sentence

2014-01-18T07:49:52Z

Rational numbers: fix awkward sentence

← Older revision		Revision as of 09:49, 18 January 2014
Line 1:		Line 1:
	~~Jayson Berryhill is how I~~'~~m known as and my spouse doesn~~'~~t like it at all~~. ~~I am really fond~~ of to ~~go to karaoke~~ but I'~~ve been using on new issues lately~~. ~~Alaska psychics online~~; [~~http~~://~~www~~.~~youronlinepublishers~~.~~com~~/~~authWiki~~/~~AdolphvhBladenqq www~~.~~youronlinepublishers~~.~~com~~], is the ~~only place I~~'~~ve been residing~~ in ~~but now I~~'~~m considering other choices~~. ~~Since I was eighteen I~~'~~ve been operating as~~ a ~~bookkeeper but quickly my wife~~ and ~~I will begin our own business~~.<br><br>~~Stop phone psychic readings~~ ([~~http~~://~~mybrandcp~~.~~com~~/xe/~~board_XmDx25~~/~~107997 http~~://~~mybrandcp~~.~~com~~/xe/~~board_XmDx25~~/~~107997~~]) ~~by my web page~~ - ~~psychic readings online~~ ([http://www.~~sirudang~~.~~com~~/~~siroo_Notice~~/~~2110 click~~ the ~~up coming web page~~])		[[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 (''[http://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]]s that responds ''collectively'' to [[electromagnetic force]]s. Plasma typically takes the form of neutral gas-like clouds or charged [[ion beam]]s, but may also include dust and grains. <ref>Peratt, Anthony, ''Physics of the Plasma Universe'' (1992); </ref> The behaviour of such particle systems can be studied statistically. <ref>Parks, George K., Physics of Space Plasmas (2004, 2nd Ed.)</ref>

			== Fundamental plasma parameters ==

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

			=== Frequencies ===
			*'''electron gyrofrequency''', the angular frequency of the circular motion of an electron in the plane perpendicular to the magnetic field:
			:<math>\omega_{ce} = eB/m_ec = 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>\omega_{ci} = eB/m_ic = 9.58 \times 10^3 Z \mu^{-1} B \mbox{rad/s} \,</math>
			*'''electron plasma frequency''', the frequency with which electrons oscillate ([[plasma oscillation]]):
			:<math>\omega_{pe} = (4\pi n_ee^2/m_e)^{1/2} = 5.64 \times 10^4 n_e^{1/2} \mbox{rad/s}</math>
			*'''ion plasma frequency''':
			:<math>\omega_{pi} = (4\pi n_iZ^2e^2/m_i)^{1/2} = 1.32 \times 10^3 Z \mu^{-1/2} n_i^{1/2} \mbox{rad/s}</math>
			*'''electron trapping rate''':
			:<math>\nu_{Te} = (eKE/m_e)^{1/2} = 7.26 \times 10^8 K^{1/2} E^{1/2} \mbox{s}^{-1} \,</math>
			*'''ion trapping rate''':
			:<math>\nu_{Ti} = (ZeKE/m_i)^{1/2} = 1.69 \times 10^7 Z^{1/2} K^{1/2} E^{1/2} \mu^{-1/2} \mbox{s}^{-1} \,</math>
			*'''electron collision rate''':
			:<math>\nu_e = 2.91 \times 10^{-6} n_e\,\ln\Lambda\,T_e^{-3/2} \mbox{s}^{-1}</math>
			*'''ion collision rate''':
			:<math>\nu_i = 4.80 \times 10^{-8} Z^4 \mu^{-1/2} n_i\,\ln\Lambda\,T_i^{-3/2} \mbox{s}^{-1}</math>

			=== Lengths ===
			*'''[[Thermal de Broglie wavelength\|Electron thermal de Broglie wavelength]]''', approximate average [[de Broglie wavelength]] of electrons in a plasma:
			:<math>\Lambda_e= \sqrt{\frac{h^2}{2\pi m_ekT_e}}= 6.919\times 10^{-8}\,T_e^{-1/2}\,\mbox{cm}</math>
			*'''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 have a velocity typical of the temperature, ignoring quantum-mechanical effects:
			:<math>e^2/kT=1.44\times10^{-7}\,T^{-1}\,\mbox{cm}</math>
			*'''electron gyroradius''', the radius of the circular motion of an electron in the plane perpendicular to the magnetic field:
			:<math>r_e = v_{Te}/\omega_{ce} = 2.38\,T_e^{1/2}B^{-1}\,\mbox{cm}</math>
			*'''ion gyroradius''', the radius of the circular motion of an ion in the plane perpendicular to the magnetic field:
			:<math>r_i = v_{Ti}/\omega_{ci} = 1.02\times10^2\,\mu^{1/2}Z^{-1}T_i^{1/2}B^{-1}\,\mbox{cm}</math>
			*'''plasma [[skin depth]]''', the depth in a plasma to which electromagnetic radiation can penetrate:
			:<math>c/\omega_{pe} = 5.31\times10^5\,n_e^{-1/2}\,\mbox{cm}</math>
			*'''[[Debye length]]''', the scale over which electric fields are screened out by a redistribution of the electrons:
			:<math>\lambda_D = (kT/4\pi ne^2)^{1/2} = 7.43\times10^2\,T^{1/2}n^{-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>d_i = c/\omega_{pi}</math>

			=== Velocities ===
			*'''electron thermal velocity''', typical velocity of an electron in a [[Maxwell-Boltzmann distribution]]:
			:<math>v_{Te} = (kT_e/m_e)^{1/2} = 4.19\times10^7\,T_e^{1/2}\,\mbox{cm/s}</math>
			*'''ion thermal velocity''', typical velocity of an ion in a [[Maxwell-Boltzmann distribution]]:
			:<math>v_{Ti} = (kT_i/m_i)^{1/2} = 9.79\times10^5\,\mu^{-1/2}T_i^{1/2}\,\mbox{cm/s}</math>
			*'''ion sound velocity''', the speed of the longitudinal waves resulting from the mass of the ions and the pressure of the electrons:
			:<math>c_s = (\gamma ZkT_e/m_i)^{1/2} = 9.79\times10^5\,(\gamma ZT_e/\mu)^{1/2}\,\mbox{cm/s}</math>
			*'''[[Hannes Alfvén\|Alfvén]] velocity''', the speed of the [[Alfvén wave\|waves]] resulting from the mass of the ions and the restoring force of the magnetic field:
			:<math>v_A = B/(4\pi n_im_i)^{1/2} = 2.18\times10^{11}\,\mu^{-1/2}n_i^{-1/2}B\,\mbox{cm/s}</math>

			=== 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.]]
			*square root of electron/proton mass ratio
			:<math>(m_e/m_p)^{1/2} = 2.33\times10^{-2} = 1/42.9 \,</math>
			* number of particles in a Debye sphere
			:<math>(4\pi/3)n\lambda_D^3 = 1.72\times10^9\,T^{3/2}n^{-1/2}</math>
			* Alfvén velocity/speed of light
			:<math>v_A/c = 7.28\,\mu^{-1/2}n_i^{-1/2}B</math>
			* electron plasma/gyrofrequency ratio
			:<math>\omega_{pe}/\omega_{ce} = 3.21\times10^{-3}\,n_e^{1/2}B^{-1}</math>
			* ion plasma/gyrofrequency ratio
			:<math>\omega_{pi}/\omega_{ci} = 0.137\,\mu^{1/2}n_i^{1/2}B^{-1}</math>
			* thermal/magnetic pressure ratio ("[[Beta (plasma physics)\|beta]]")
			:<math>\beta = 8\pi nkT/B^2 = 4.03\times10^{-11}\,nTB^{-2}</math>
			* magnetic/ion rest energy ratio
			:<math>B^2/8\pi n_im_ic^2 = 26.5\,\mu^{-1}n_i^{-1}B^2</math>

			==See also==
			*[[List of plasma (physics) articles]]

			==References==
			* [http://www.ipp.mpg.de/~dpc/nrl/ NRL Plasma Formulary] (esp. [http://www.ipp.mpg.de/~dpc/nrl/28.html p. 28] and [http://www.ipp.mpg.de/~dpc/nrl/29.html p. 29]), J.D. Huba, [[United States Naval Research Laboratory\|Naval Research Laboratory]] (2007)

			==Footnotes==
			<div style="font-size: 95%">
			<!--See [[Wikipedia:Footnotes]] for an explanation of how to generate footnotes using the <ref(erences/)> tags-->
			<references/>
			</div>

			<!--Categories-->
			[[Category:Plasma physics\| ]]
			[[Category:Plasma diagnostics]]
			[[Category:Astrophysics]]

en>Pjoef: Disambiguated: Canonical#Mathematics → canonical form

2012-03-03T10:52:05Z

Disambiguated: Canonical#Mathematics → canonical form

New page

Jayson Berryhill is how I'm known as and my spouse doesn't like it at all. I am really fond of to go to karaoke but I've been using on new issues lately. Alaska psychics online; [http://www.youronlinepublishers.com/authWiki/AdolphvhBladenqq www.youronlinepublishers.com], is the only place I've been residing in but now I'm considering other choices. Since I was eighteen I've been operating as a bookkeeper but quickly my wife and I will begin our own business.<br><br>Stop phone psychic readings ([http://mybrandcp.com/xe/board_XmDx25/107997 http://mybrandcp.com/xe/board_XmDx25/107997]) by my web page - psychic readings online ([http://www.sirudang.com/siroo_Notice/2110 click the up coming web page])