Bijective numeration - Revision history

en>Vadmium: /* top */ Reference article on bases, not a particular base called k

2015-01-08T03:26:48Z

top: Reference article on bases, not a particular base called k

← Older revision		Revision as of 05:26, 8 January 2015
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	~~Andera~~ is ~~what you can contact her but she by no means really favored that title~~. ~~My working day~~ occupation is ~~an invoicing officer~~ but ~~I've currently utilized~~ for ~~another one~~. ~~To climb~~ is ~~some thing she would never give up. Mississippi is where~~ her ~~home is~~ but ~~her husband wants them~~ to ~~move~~.<br><br>~~Here is~~ my ~~web-site~~: [http://~~isaworld~~.Pe.~~kr/~~?~~document_srl~~=~~392088 best psychic readings~~]		The author is called Irwin. Hiring is his occupation. One of the issues he loves most is ice skating but he is having difficulties to discover time for it. North Dakota is her beginning place but she will have to transfer one day or another.<br><br>Check out my page :: [http://nemoonehfilm.ir/index.php?do=/profile-6876/info/ nemoonehfilm.ir]

en>Mark viking: /* Definition */ dummy edit to force server purge

2014-03-01T02:01:00Z

Definition: dummy edit to force server purge

← Older revision		Revision as of 04:01, 1 March 2014
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	~~In [[atomic physics]], '''Doppler broadening'''~~ is ~~the broadening of [[spectral line]]s due to the [[Doppler effect]] caused~~ by a distribution of velocities of [[atom]]s or [[molecule]]s. Different velocities of the [[Spontaneous_emission\|emitting]] particles result in different (Doppler) shifts, the cumulative effect of which is the line broadening.<ref>{{cite book		Andera is what you can contact her but she by no means really favored that title. My working day occupation is an invoicing officer but I've currently utilized for another one. To climb is some thing she would never give up. Mississippi is where her home is but her husband wants them to move.<br><br>Here is my web-site: [http://isaworld.Pe.kr/?document_srl=392088 best psychic readings]
	\|title~~=Lasers~~
	~~\|author=Siegman, AE~~
	~~\|year=1986~~
	~~\|url=http://books~~.~~google.dk/books?id=1BZVwUZLTkAC&lpg=PA1184&ots=6xdm1N2jLf&dq=doppler%20broadening%20Siegman&hl=en&pg=PA1184#v=onepage&q=doppler%20broadening%20Siegman&f=false~~
	~~}}</ref>~~
	~~This resulting line profile~~ is ~~known as a '''Doppler profile'''. A particular and perhaps the most important case is the~~ '''thermal Doppler broadening''' due to the [[Kinetic_theory\|thermal motion]] of the particles. Then, the broadening depends only on the [[frequency]] of the spectral line, the [[mass]] of the emitting particles, and their [[temperature]], and therefore can be used for ~~inferring the temperature of an emitting body~~.

	[[Saturated absorption spectroscopy]], also known as Doppler-free spectroscopy, can be used to find the true frequency of an atomic transition without cooling a sample down to temperatures at which the Doppler broadening is ~~minimal~~.

	~~== Derivation ==~~

	When thermal motion causes a particle to move towards the observer, the emitted radiation will be shifted to a higher frequency. Likewise, when the emitter moves away, the frequency will be lowered. For non-relativistic thermal velocities, the [[Doppler effect\|Doppler shift]] in frequency will be:

	~~:<math>f = f_0\left(1+\frac{v}{c}\right)</math>~~

	where <math>\ f</math> is the observed frequency, <math>\ f_0</math> is the rest frequency, <math>\ v</math> is the velocity of the emitter towards the observer, and <math>c</math> is the [[speed of light]].

	Since there is a distribution of speeds both toward and away from the observer in any volume element of the radiating body, the net effect will be to broaden the observed line. If <math>\,P_v(v)dv</math> is the fraction of particles with velocity component <math>\,v</math> to <math>\,v+dv</math> along a line of sight, then the corresponding distribution of the frequencies is

	~~:<math>P_f(f)df = P_v(v_f)\frac{dv}{df}df</math>,~~

	where ~~<math>\,v_f = c\left(\frac{f}{f_0} - 1\right)</math>~~ is ~~the velocity towards the observer corresponding~~ to ~~the shift of the rest frequency <math>\,f_0</math> to <math>\,f</math>~~. ~~Therefore,~~

	~~::{\|cellpadding="2" style="border:2px solid #ccccff"~~
	\|<~~math~~>~~P_f(f)df = \frac{c}{f_0}P_v\left(c\left(\frac{f}{f_0} - 1\right)\right)df~~<~~/math>.~~
	\|}

	We can also express the broadening in terms of the [[wavelength]] <math>\,\lambda</math>. Recalling that in the non-relativistic limit <math>\frac{\lambda-\lambda_{0}}{\lambda_{0}} \approx -\frac{f-f_0}{f_0}</math>, we obtain

	~~::{\|cellpadding="2" style="border:2px solid #ccccff"~~
	~~\|<math>P_\lambda(\lambda)d\lambda = \frac{c}{\lambda_0}P_v\left(c\left(1 - \frac{\lambda}{\lambda_0}\right)\right)d\lambda</math>.~~
	\|}

	~~In the case of the thermal Doppler broadening, the velocity distribution is given by the [[Maxwell distribution]]~~
	~~:<math>P_v(v)dv = \sqrt{\frac{m}{2\pi kT}}\,\exp\left(-\frac{mv^2}{2kT}\right)dv</math>,~~

	~~where <math>\,m</math> is the mass of the emitting particle, <math>\,T</math~~> is ~~the temperature and <math>\,k</math> is the [[Boltzmann constant]].~~

	~~Then,~~

	~~:<math>P_f(f)df=\left(\frac{c}{f_0}\right)\sqrt{\frac{m}{2\pi kT}}\,\exp\left(-\frac{m\left[c\left(\frac{f}{f_0}~~-~~1\right)\right]^2}{2kT}\right)df</math>.~~

	~~We can simplify this expression as~~

	:~~<math>P_f(f)df=\sqrt{\frac{mc^2}{2\pi kT {f_0}^2}}\,~~
	~~\exp\left(-\frac{mc^2\left(f-f_0\right)^2}{2kT {f_0}^2}\right)df</math>,~~

	~~which we immediately recognize as a~~ [~~[Gaussian_function\|Gaussian profile]] with the [[standard deviation]]~~

	:~~<math>\sigma_{f} = \sqrt{\frac{kT}{mc^2}}f_0<~~/~~math>~~

	~~and [[full width at half maximum]] (FWHM)~~

	~~::{\|cellpadding="2" style="border:2px solid #ccccff"~~
	~~\|<math>\Delta f_{\text{FWHM}} = \sqrt{\frac{8kT\ln 2}{mc^2}}f_{0}<~~/~~math>.~~
	\|}

	~~== Applications and caveats ==~~

	In [[astronomy]] and [[plasma physics]], the thermal Doppler broadening is one of the explanations for the broadening of spectral lines, and as such gives an indication for the temperature of observed material. It should be noted, though, that other causes of velocity distributions may exist, e.~~g., due to [[turbulence\|turbulent]] motion~~. ~~For a fully developed turbulence, the resulting line profile is generally very difficult to distinguish from the thermal one.<ref>{{cite book~~
	~~\| first = Hans R. \| last = Griem \| year = 1997~~
	~~\| title = Principles of Plasmas Spectroscopy~~
	~~\| publisher = University Press \| location = Cambridge \| isbn = 0-521-45504-9 }}<~~/~~ref>~~
	Another cause could be a large range of ''macroscopic'' velocities resulting, e.g., from the receding and approaching portions of a rapidly spinning [[accretion disk]]. Finally, there are many other factors which can also broaden the lines. For example, a sufficiently high particle [[number density]] may lead to significant [[Stark broadening]].

	~~==See also==~~
	* [[Mössbauer effect]]
	* [[Dicke effect]]

	=~~=References==~~
	~~<references/>~~

	~~[[Category:Doppler effects]]~~
	~~[[Category:Physical phenomena]~~]

en>SyaWgnignahCehT: /* The bijective base-26 system */ Redundancy removed

2013-11-08T11:50:03Z

The bijective base-26 system: Redundancy removed

← Older revision		Revision as of 13:50, 8 November 2013
Line 1:		Line 1:
	~~Let me first start~~ by ~~introducing myself~~. My title is ~~Boyd Butts even though it~~ is ~~not~~ the ~~title~~ on ~~my birth certificate~~. ~~North Dakota~~ is ~~her birth place but she~~ will ~~have~~ to ~~transfer one day or another~~. ~~To gather coins~~ is ~~what her family~~ and ~~her enjoy~~. ~~Hiring~~ is ~~his occupation~~.<br><br>~~my site~~ :: ~~over~~ the ~~counter std test~~ ([~~http~~://~~www~~.1a-~~pornotube~~.~~com~~/~~blog~~/~~84958 simply click~~ the ~~following website page~~])		In [[atomic physics]], '''Doppler broadening''' is the broadening of [[spectral line]]s due to the [[Doppler effect]] caused by a distribution of velocities of [[atom]]s or [[molecule]]s. Different velocities of the [[Spontaneous_emission\|emitting]] particles result in different (Doppler) shifts, the cumulative effect of which is the line broadening.<ref>{{cite book
			\|title=Lasers
			\|author=Siegman, AE
			\|year=1986
			\|url=http://books.google.dk/books?id=1BZVwUZLTkAC&lpg=PA1184&ots=6xdm1N2jLf&dq=doppler%20broadening%20Siegman&hl=en&pg=PA1184#v=onepage&q=doppler%20broadening%20Siegman&f=false
			}}</ref>
			This resulting line profile is known as a '''Doppler profile'''. A particular and perhaps the most important case is the '''thermal Doppler broadening''' due to the [[Kinetic_theory\|thermal motion]] of the particles. Then, the broadening depends only on the [[frequency]] of the spectral line, the [[mass]] of the emitting particles, and their [[temperature]], and therefore can be used for inferring the temperature of an emitting body.

			[[Saturated absorption spectroscopy]], also known as Doppler-free spectroscopy, can be used to find the true frequency of an atomic transition without cooling a sample down to temperatures at which the Doppler broadening is minimal.

			== Derivation ==

			When thermal motion causes a particle to move towards the observer, the emitted radiation will be shifted to a higher frequency. Likewise, when the emitter moves away, the frequency will be lowered. For non-relativistic thermal velocities, the [[Doppler effect\|Doppler shift]] in frequency will be:

			:<math>f = f_0\left(1+\frac{v}{c}\right)</math>

			where <math>\ f</math> is the observed frequency, <math>\ f_0</math> is the rest frequency, <math>\ v</math> is the velocity of the emitter towards the observer, and <math>c</math> is the [[speed of light]].

			Since there is a distribution of speeds both toward and away from the observer in any volume element of the radiating body, the net effect will be to broaden the observed line. If <math>\,P_v(v)dv</math> is the fraction of particles with velocity component <math>\,v</math> to <math>\,v+dv</math> along a line of sight, then the corresponding distribution of the frequencies is

			:<math>P_f(f)df = P_v(v_f)\frac{dv}{df}df</math>,

			where <math>\,v_f = c\left(\frac{f}{f_0} - 1\right)</math> is the velocity towards the observer corresponding to the shift of the rest frequency <math>\,f_0</math> to <math>\,f</math>. Therefore,

			::{\|cellpadding="2" style="border:2px solid #ccccff"
			\|<math>P_f(f)df = \frac{c}{f_0}P_v\left(c\left(\frac{f}{f_0} - 1\right)\right)df</math>.
			\|}

			We can also express the broadening in terms of the [[wavelength]] <math>\,\lambda</math>. Recalling that in the non-relativistic limit <math>\frac{\lambda-\lambda_{0}}{\lambda_{0}} \approx -\frac{f-f_0}{f_0}</math>, we obtain

			::{\|cellpadding="2" style="border:2px solid #ccccff"
			\|<math>P_\lambda(\lambda)d\lambda = \frac{c}{\lambda_0}P_v\left(c\left(1 - \frac{\lambda}{\lambda_0}\right)\right)d\lambda</math>.
			\|}

			In the case of the thermal Doppler broadening, the velocity distribution is given by the [[Maxwell distribution]]
			:<math>P_v(v)dv = \sqrt{\frac{m}{2\pi kT}}\,\exp\left(-\frac{mv^2}{2kT}\right)dv</math>,

			where <math>\,m</math> is the mass of the emitting particle, <math>\,T</math> is the temperature and <math>\,k</math> is the [[Boltzmann constant]].

			Then,

			:<math>P_f(f)df=\left(\frac{c}{f_0}\right)\sqrt{\frac{m}{2\pi kT}}\,\exp\left(-\frac{m\left[c\left(\frac{f}{f_0}-1\right)\right]^2}{2kT}\right)df</math>.

			We can simplify this expression as

			:<math>P_f(f)df=\sqrt{\frac{mc^2}{2\pi kT {f_0}^2}}\,
			\exp\left(-\frac{mc^2\left(f-f_0\right)^2}{2kT {f_0}^2}\right)df</math>,

			which we immediately recognize as a [[Gaussian_function\|Gaussian profile]] with the [[standard deviation]]

			:<math>\sigma_{f} = \sqrt{\frac{kT}{mc^2}}f_0</math>

			and [[full width at half maximum]] (FWHM)

			::{\|cellpadding="2" style="border:2px solid #ccccff"
			\|<math>\Delta f_{\text{FWHM}} = \sqrt{\frac{8kT\ln 2}{mc^2}}f_{0}</math>.
			\|}

			== Applications and caveats ==

			In [[astronomy]] and [[plasma physics]], the thermal Doppler broadening is one of the explanations for the broadening of spectral lines, and as such gives an indication for the temperature of observed material. It should be noted, though, that other causes of velocity distributions may exist, e.g., due to [[turbulence\|turbulent]] motion. For a fully developed turbulence, the resulting line profile is generally very difficult to distinguish from the thermal one.<ref>{{cite book
			\| first = Hans R. \| last = Griem \| year = 1997
			\| title = Principles of Plasmas Spectroscopy
			\| publisher = University Press \| location = Cambridge \| isbn = 0-521-45504-9 }}</ref>
			Another cause could be a large range of ''macroscopic'' velocities resulting, e.g., from the receding and approaching portions of a rapidly spinning [[accretion disk]]. Finally, there are many other factors which can also broaden the lines. For example, a sufficiently high particle [[number density]] may lead to significant [[Stark broadening]].

			==See also==
			* [[Mössbauer effect]]
			* [[Dicke effect]]

			==References==
			<references/>

			[[Category:Doppler effects]]
			[[Category:Physical phenomena]]

216.75.227.188: Added reference to Greek numeral system

2012-03-23T01:50:03Z

Added reference to Greek numeral system

New page

Let me first start by introducing myself. My title is Boyd Butts even though it is not the title on my birth certificate. North Dakota is her birth place but she will have to transfer one day or another. To gather coins is what her family and her enjoy. Hiring is his occupation.<br><br>my site :: over the counter std test ([http://www.1a-pornotube.com/blog/84958 simply click the following website page])