Total factor productivity - Revision history

en>Bender235 at 16:44, 11 January 2015

2015-01-11T16:44:42Z

← Older revision		Revision as of 18:44, 11 January 2015
Line 1:		Line 1:
	~~In [[plasma physics]], an '''ion acoustic wave''' is one type of [[longitudinal wave\|longitudinal]] oscillation of~~ the ~~[[ion]]s~~ and ~~[[electron]]s in a [[plasma (physics)\|plasma]],~~ much like [[acoustic wave]]s traveling in neutral gas. However, because the waves propagate through positively charged ions, ion acoustic waves can interact with their [[electromagnetic field]]s, as well as simple collisions. In plasmas, ion acoustic waves are frequently referred to as acoustic waves or even just sound waves. They commonly govern the evolution of mass density, for ~~instance due to~~ [[pressure gradient]]s, on time scales longer than the frequency corresponding to the relevant length scale. Ion acoustic waves can occur in an unmagnetized plasma or in a magnetized plasma parallel to the [[magnetic field]]. For a single species plasma, the waves are [[Dispersion relation\|dispersionless]] (<math>\omega=v_sk</math>) with a speed (in the long [[wavelength]] limit) given by {{Citation needed\|date=October 2013}}		Another day I woke up and realized - I have also been solitary for a little while at the moment and following much intimidation from pals I today find myself opted for [http://www.adobe.com/cfusion/search/index.cfm?term=&internet+dating&loc=en_us&siteSection=home internet dating]. They guaranteed me that [http://www.banburycrossonline.com where to buy luke bryan tickets] there are plenty of fun, sweet and regular folks to meet, therefore here goes the toss!<br>I attempt to keep as physically healthy as possible staying at the gym many times per week. I appreciate my sports and make an effort to luke bryan where is he from ([http://lukebryantickets.iczmpbangladesh.org moved here]) perform or view as several a possible. Being winter I shall frequently at Hawthorn fits. Note: Supposing that you will contemplated shopping an athletics I do not mind, I have experienced the carnage of wrestling matches at stocktake sales.<br>My family [http://www.senatorwonderling.com luke bryan winter tour] and buddies are amazing and spending some time together at tavern gigs or dishes is consistently essential. As I realize you could never own a significant dialogue using the noise I haven't ever been into night clubs. Additionally, I have two unquestionably cheeky and quite cunning puppies that are always enthusiastic to meet up fresh people.<br><br>Feel free [http://lukebryantickets.lazintechnologies.com luke bryan concert tickets cheap] to surf to my website :: [http://lukebryantickets.neodga.com justin bieber concert tickets]
	:~~<math>v_s = \sqrt{\frac{\gamma_{e}ZK_{B}T_e+\gamma_{i}K_{B}T_i}{M}}<~~/~~math>~~
	~~where <math>K_{B}<~~/~~math> is [[Boltzmann's constant]], <math>M<~~/~~math> is the mass of the ion, <math>Z<~~/~~math> is its charge, <math>T_e<~~/~~math> is the temperature of the electrons and <math>T_i</math> is the temperature of the ions~~. Normally γ<sub>e</sub> is taken to be unity, on the grounds that the [[thermal conductivity]] of electrons is large enough to keep them [[isothermal]] on the time scale of ion acoustic waves, and γ<sub>i</sub> is taken to be 3, corresponding to one-dimensional motion. In [[collisionless]] plasmas, the electrons are often much hotter than the ions, in which case the second term ~~in the numerator can be ignored.~~

	== ~~Derivation~~ ==
	~~We derive the ion acoustic wave dispersion relation for a linearized fluid description of a plasma with multiple ion species~~. ~~A subscript 0 denotes constant equilibrium quantities, and 1 denotes first-order perturbations. We assume the pressure perturbations for each species (electrons and ions) are a~~ [~~[Polytropic_process]], namely <math>p_{s1} = \gamma_s T_{s0} n_{s1}</math> for species s. Using the ion continuity equation, the ion momentum equation becomes~~
	:~~<math>(-m_i\partial_{tt}+\gamma_iT_i\nabla^2)n_{i1} = Z_ien_{i0}\nabla\cdot\vec E </math>~~
	~~We relate the electric field <math>\vec E_1<~~/~~math> to the electron density by the electron momentum equation:~~
	~~:<math>n_{e0}m_e\partial_t\vec v_{e1} = -n_{e0}e\vec E_1 - \gamma_eT_e\nabla n_{e1} <~~/~~math>~~
	~~We now neglect the left-hand side, which is due to electron inertia~~. ~~This is valid for waves with frequencies much less than the electron plasma frequency~~. ~~The resulting electric field is~~
	~~:<math>\vec E_1 = - {\gamma_eT_e \over n_{e0}e}\nabla n_{e1} </math>~~
	~~Since we have already solved for the electric field, we cannot also find it from Poisson's equation. The ion momentum equation now relates <math>n_{i1}</math> for each species~~ to ~~<math>n_{e1}</math>:~~
	~~:<math>(-m_i\partial_{tt}+\gamma_iT_i\nabla^2)n_{i1} = -\gamma_e T_e \nabla^2 n_{e1} </math>~~
	~~We arrive at a dispersion relation via Poisson's equation:~~
	~~:<math> {\epsilon_0 \over e}\nabla\cdot\vec E = [\sum_i n_{i0}Z_i - n_{ne0}~~] ~~+ [\sum_in_{i1}Z_i - n_{e1}] </math>~~
	~~The first bracketed term on the right is zero by assumption (charge-neutral equilibrium). We substitute for the electric field~~ and ~~rearrange~~ to ~~find~~
	~~:<math> (1-\gamma_e \lambda_{De}^2\nabla^2)n_{e1} = \sum_iZ_in_{i1} </math>.~~
	~~<math>\lambda_{De}^2 \equiv \epsilon_0T_e/(n_{e0}e^2)</math> defines~~ the ~~electron Debye length. The second term on the left arises from the <math>\nabla\cdot\vec E~~<~~/math~~> ~~term, and reflects the degree~~ to ~~which the perturbation is not charge-neutral. If <math>k\lambda_{De}</math> is small we may drop this term. This approximation is sometimes called~~ the ~~plasma approximation~~.

	~~We now work in Fourier space,~~ and ~~find~~
	~~:<math> n_{i1} = \gamma_eT_eZ_i {n_{i0} \over n_{e0}} [m_iv_s^2-\gamma_iT_i]^{-1} n_{e1} </math>~~
	~~<math>v_s=\omega/k</math> is the wave phase velocity. Substituting this into Poisson's equation gives us~~ an ~~expression~~ where ~~each term~~ is ~~proportional to <math>n_{e1}<~~/~~math>~~. ~~To find the dispersion relation for natural modes, we look for solutions for <math>n_{e1}</math> nonzero~~.
	~~:<math> \gamma_eT_e \sum_i Z_i^2f_i[m_iv_s^2-\gamma_iT_i~~]~~^{-1} = \bar Z(1+\gamma_ek^2\lambda_{De}^2~~)~~</math>~~

	~~<math>n_{i1}=f_in_{I1}</math> where <math>n_{I1}=\Sigma_i n_{i1}</math>, and <math>\bar Z = \Sigma_i Z_if_i </math>~~. ~~In general it is~~ not ~~possible to further simplify this expression. If <math>k\lambda_{De}</math> is small (the plasma approximation)~~, ~~we can neglect the second term on the term, and~~ the ~~wave is dispersionless <math>\omega = v_sk</math> with <math>v_s</math> independent~~ of k.

	~~== Specific Examples ==~~
	~~To illustrate some features of ion acoustic waves, we can consider some specific examples of the general dispersion relation given above~~. ~~First, for a single ion species, we find~~
	:<~~math~~>~~v_s^2 = {\gamma_eZ_iT_e \over m_i}~~[~~{1 \over 1+\gamma_e(k\lambda_{De})^2} + {\gamma_iT_{i0} \over \gamma_eZ_iT_{e0}}]~~
	~~</math>~~

	~~For any number of ion species, all of which are cold (<math>\gamma_iT_{i0} \ll m_iv_s^2</math>), we obtain~~
	:~~<math>v_s^2 = {\gamma_eT_{e0} \over 1+\gamma_e(k\lambda_{De})^2} \sum_i {Z_i^2f_i \over \bar Z m_i}</math>~~

	~~A case of interest to nuclear fusion is an equimolar mixture of deuterium and tritium ions (<math>f_D=f_T=1~~/2</~~math>)~~. ~~Let us specialize to full ionization (<math>Z_D=Z_T=1</math>), equal temperatures (<math>T_{e0}=T_{i0}\equiv T_0</math>), polytrope exponents <math>\gamma_e=1, \gamma_i=3</math>,~~ and ~~neglect the <math>(k\lambda_{De})^2</math> contribution~~. ~~The dispersion relation becomes~~ a ~~quadratic in <math>v_s^2</math>, namely:~~
	~~:<math>2A_DA_Tu^4 - 7(A_D+A_T)u^2 + 24=0</math>~~
	~~where <math>A_i=m_i/m_u</math>, <math>m_u</math> is~~ the ~~atomic mass unit, and <math>u^2=m_uv_s^2/T_0</math>~~. ~~Using <math>(A_D~~,~~A_T)=(2.01,3.02)</math> we find the~~ two ~~roots~~ are <~~math~~>~~u^2=(1.10,1.81)~~<~~/math~~>.

	~~== Damping ==~~
	~~Ion acoustic waves are damped both by~~ [~~[Coulomb collision]]s and collisionless [[Landau damping]]~~. ~~The Landau damping occurs on both electrons and ions, with the relative importance depending on parameters~~.

	~~==See also==~~
	* [[Waves in plasmas]]
	* [[Sound]]
	* [[Alfvén wave]]
	* [[Magnetosonic wave]]
	* [[List of plasma (physics) articles]]

	~~==External links==~~
	*[http://~~mr-fusion~~.~~hellblazer~~.com~~/pdfs/ion-accoustic-waves-and-langmuir-waves.pdf Various patents and articles related to fusion, IEC, ICC and plasma physics]~~

	~~{{DEFAULTSORT:Ion Acoustic Wave}}~~
	~~[[Category:Waves in plasmas]~~]

en>ChrisGualtieri: Remove stub template(s). Page is start class or higher. Also check for and do General Fixes + Checkwiki fixes using AWB

2013-12-12T23:31:34Z

Remove stub template(s). Page is start class or higher. Also check for and do General Fixes + Checkwiki fixes using AWB

← Older revision		Revision as of 01:31, 13 December 2013
Line 1:		Line 1:
	~~My name~~ is ~~Ernie Burch but everybody calls me Ernie~~. I'm from ~~United States~~. I'~~m studying at~~ the ~~college~~ (~~2nd year~~) and ~~I play~~ the ~~Pedal Steel Guitar~~ for ~~6 years~~. ~~Usually I choose music from my famous films ;~~). <br>~~I have two sister~~. ~~I like Basket Weaving~~, ~~watching TV~~ (~~Sons~~ of ~~Anarchy~~) and ~~Record collecting~~.<br><br>~~Here~~ is ~~my blog~~ ... [http://~~spmba~~.de/~~index~~.~~php?mod=users&action=view&id=16685 fifa coin Generator~~]		In [[plasma physics]], an '''ion acoustic wave''' is one type of [[longitudinal wave\|longitudinal]] oscillation of the [[ion]]s and [[electron]]s in a [[plasma (physics)\|plasma]], much like [[acoustic wave]]s traveling in neutral gas. However, because the waves propagate through positively charged ions, ion acoustic waves can interact with their [[electromagnetic field]]s, as well as simple collisions. In plasmas, ion acoustic waves are frequently referred to as acoustic waves or even just sound waves. They commonly govern the evolution of mass density, for instance due to [[pressure gradient]]s, on time scales longer than the frequency corresponding to the relevant length scale. Ion acoustic waves can occur in an unmagnetized plasma or in a magnetized plasma parallel to the [[magnetic field]]. For a single species plasma, the waves are [[Dispersion relation\|dispersionless]] (<math>\omega=v_sk</math>) with a speed (in the long [[wavelength]] limit) given by {{Citation needed\|date=October 2013}}
			:<math>v_s = \sqrt{\frac{\gamma_{e}ZK_{B}T_e+\gamma_{i}K_{B}T_i}{M}}</math>
			where <math>K_{B}</math> is [[Boltzmann's constant]], <math>M</math> is the mass of the ion, <math>Z</math> is its charge, <math>T_e</math> is the temperature of the electrons and <math>T_i</math> is the temperature of the ions. Normally γ<sub>e</sub> is taken to be unity, on the grounds that the [[thermal conductivity]] of electrons is large enough to keep them [[isothermal]] on the time scale of ion acoustic waves, and γ<sub>i</sub> is taken to be 3, corresponding to one-dimensional motion. In [[collisionless]] plasmas, the electrons are often much hotter than the ions, in which case the second term in the numerator can be ignored.

			== Derivation ==
			We derive the ion acoustic wave dispersion relation for a linearized fluid description of a plasma with multiple ion species. A subscript 0 denotes constant equilibrium quantities, and 1 denotes first-order perturbations. We assume the pressure perturbations for each species (electrons and ions) are a [[Polytropic_process]], namely <math>p_{s1} = \gamma_s T_{s0} n_{s1}</math> for species s. Using the ion continuity equation, the ion momentum equation becomes
			:<math>(-m_i\partial_{tt}+\gamma_iT_i\nabla^2)n_{i1} = Z_ien_{i0}\nabla\cdot\vec E </math>
			We relate the electric field <math>\vec E_1</math> to the electron density by the electron momentum equation:
			:<math>n_{e0}m_e\partial_t\vec v_{e1} = -n_{e0}e\vec E_1 - \gamma_eT_e\nabla n_{e1} </math>
			We now neglect the left-hand side, which is due to electron inertia. This is valid for waves with frequencies much less than the electron plasma frequency. The resulting electric field is
			:<math>\vec E_1 = - {\gamma_eT_e \over n_{e0}e}\nabla n_{e1} </math>
			Since we have already solved for the electric field, we cannot also find it from Poisson's equation. The ion momentum equation now relates <math>n_{i1}</math> for each species to <math>n_{e1}</math>:
			:<math>(-m_i\partial_{tt}+\gamma_iT_i\nabla^2)n_{i1} = -\gamma_e T_e \nabla^2 n_{e1} </math>
			We arrive at a dispersion relation via Poisson's equation:
			:<math> {\epsilon_0 \over e}\nabla\cdot\vec E = [\sum_i n_{i0}Z_i - n_{ne0}] + [\sum_in_{i1}Z_i - n_{e1}] </math>
			The first bracketed term on the right is zero by assumption (charge-neutral equilibrium). We substitute for the electric field and rearrange to find
			:<math> (1-\gamma_e \lambda_{De}^2\nabla^2)n_{e1} = \sum_iZ_in_{i1} </math>.
			<math>\lambda_{De}^2 \equiv \epsilon_0T_e/(n_{e0}e^2)</math> defines the electron Debye length. The second term on the left arises from the <math>\nabla\cdot\vec E</math> term, and reflects the degree to which the perturbation is not charge-neutral. If <math>k\lambda_{De}</math> is small we may drop this term. This approximation is sometimes called the plasma approximation.

			We now work in Fourier space, and find
			:<math> n_{i1} = \gamma_eT_eZ_i {n_{i0} \over n_{e0}} [m_iv_s^2-\gamma_iT_i]^{-1} n_{e1} </math>
			<math>v_s=\omega/k</math> is the wave phase velocity. Substituting this into Poisson's equation gives us an expression where each term is proportional to <math>n_{e1}</math>. To find the dispersion relation for natural modes, we look for solutions for <math>n_{e1}</math> nonzero.
			:<math> \gamma_eT_e \sum_i Z_i^2f_i[m_iv_s^2-\gamma_iT_i]^{-1} = \bar Z(1+\gamma_ek^2\lambda_{De}^2)</math>

			<math>n_{i1}=f_in_{I1}</math> where <math>n_{I1}=\Sigma_i n_{i1}</math>, and <math>\bar Z = \Sigma_i Z_if_i </math>. In general it is not possible to further simplify this expression. If <math>k\lambda_{De}</math> is small (the plasma approximation), we can neglect the second term on the term, and the wave is dispersionless <math>\omega = v_sk</math> with <math>v_s</math> independent of k.

			== Specific Examples ==
			To illustrate some features of ion acoustic waves, we can consider some specific examples of the general dispersion relation given above. First, for a single ion species, we find
			:<math>v_s^2 = {\gamma_eZ_iT_e \over m_i}[{1 \over 1+\gamma_e(k\lambda_{De})^2} + {\gamma_iT_{i0} \over \gamma_eZ_iT_{e0}}]
			</math>

			For any number of ion species, all of which are cold (<math>\gamma_iT_{i0} \ll m_iv_s^2</math>), we obtain
			:<math>v_s^2 = {\gamma_eT_{e0} \over 1+\gamma_e(k\lambda_{De})^2} \sum_i {Z_i^2f_i \over \bar Z m_i}</math>

			A case of interest to nuclear fusion is an equimolar mixture of deuterium and tritium ions (<math>f_D=f_T=1/2</math>). Let us specialize to full ionization (<math>Z_D=Z_T=1</math>), equal temperatures (<math>T_{e0}=T_{i0}\equiv T_0</math>), polytrope exponents <math>\gamma_e=1, \gamma_i=3</math>, and neglect the <math>(k\lambda_{De})^2</math> contribution. The dispersion relation becomes a quadratic in <math>v_s^2</math>, namely:
			:<math>2A_DA_Tu^4 - 7(A_D+A_T)u^2 + 24=0</math>
			where <math>A_i=m_i/m_u</math>, <math>m_u</math> is the atomic mass unit, and <math>u^2=m_uv_s^2/T_0</math>. Using <math>(A_D,A_T)=(2.01,3.02)</math> we find the two roots are <math>u^2=(1.10,1.81)</math>.

			== Damping ==
			Ion acoustic waves are damped both by [[Coulomb collision]]s and collisionless [[Landau damping]]. The Landau damping occurs on both electrons and ions, with the relative importance depending on parameters.

			==See also==
			* [[Waves in plasmas]]
			* [[Sound]]
			* [[Alfvén wave]]
			* [[Magnetosonic wave]]
			* [[List of plasma (physics) articles]]

			==External links==
			*[http://mr-fusion.hellblazer.com/pdfs/ion-accoustic-waves-and-langmuir-waves.pdf Various patents and articles related to fusion, IEC, ICC and plasma physics]

			{{DEFAULTSORT:Ion Acoustic Wave}}
			[[Category:Waves in plasmas]]

en>Phmoreno: also called ''multi-factor productivity''

2012-08-12T22:06:17Z

also called ''multi-factor productivity''

New page

My name is Ernie Burch but everybody calls me Ernie. I'm from United States. I'm studying at the college (2nd year) and I play the Pedal Steel Guitar for 6 years. Usually I choose music from my famous films ;). <br>I have two sister. I like Basket Weaving, watching TV (Sons of Anarchy) and Record collecting.<br><br>Here is my blog ... [http://spmba.de/index.php?mod=users&action=view&id=16685 fifa coin Generator]