Saddle-node bifurcation - Revision history

en>V madhu: Found a typo in the marginal stability condition. The partial has to be with respect to x, and not r.

2014-10-03T16:02:05Z

Found a typo in the marginal stability condition. The partial has to be with respect to x, and not r.

← Older revision		Revision as of 18:02, 3 October 2014
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	~~[[File:Vonlaue~~.~~png\|thumb\|300px\|Ray diagram of Von Laue formulation]]~~		Hello. Let me introduce the author. Her title is Refugia Shryock. What I love doing is taking part in baseball but I haven't produced a dime with it. I am a meter reader. For a while she's been in South Dakota.<br><br>my homepage :: [http://sex-Porn-tube.ch/blog/36870 sex-Porn-tube.ch]

	~~In [[physics]], a '''Bragg plane'''~~ is ~~a [[Plane (geometry)\|plane]] in [[reciprocal space]] which bisects one reciprocal lattice vector <math>\mathbf{K}</math>~~.~~<ref>{{Cite book~~
	~~\| last1 = Ashcroft \| first1 = Neil W.~~
	~~\| last2 = Mermin \| first2 = David~~
	~~\| title = Solid State Physics~~
	~~\| publisher = Brooks Cole~~
	~~\| edition = 1~~
	~~\| date = January 2, 1976~~
	~~\| pages = 96–100~~
	~~\| isbn = 0-03-083993-9}}</ref> It~~ is ~~relevant to define this plane as~~ part ~~of the definition of the Von Laue condition for [[Interference (wave propagation)\|diffraction peaks]]~~ in ~~[[X-ray_crystallography\|x-ray diffraction crystallography]].~~

	~~Considering the diagram at right, the arriving [[x-ray]] [[plane wave]] is defined by:~~

	~~:<math>e^{i\mathbf{k}\cdot\mathbf{r}}=\cos {(\mathbf{k}\cdot\mathbf{r})} +i\sin {(\mathbf{k}\cdot\mathbf{r})}</math>~~

	~~Where <math>\mathbf{k}</math> is the incident wave vector given by:~~

	~~:<math>\mathbf{k}=\frac{2\pi}{\lambda}\hat n</math>~~

	~~where <math>\lambda</math> is the [[wavelength]] of the incident [[photon]]. While the [[Bragg~~'~~s law\|Bragg formulation]] assumes~~ a ~~unique choice of direct lattice planes and [[specular reflection]] of the incident X-rays, the Von Laue formula only assumes monochromatic light and that each scattering center acts as~~ a ~~source of secondary wavelets as described by the [[Huygens principle]]~~. ~~Each scattered wave contributes to~~ a ~~new plane wave given by:~~

	~~:<math>\mathbf{k^\prime}=\frac{2\pi}{\lambda}\hat n^\prime</math>~~

	The condition for constructive interference in the <math>\hat n^\prime</math> direction is that the path difference between the photons is an integer multiple (m) of their wavelength. We know then that for constructive interference we have:

	~~:<math>\|\mathbf{d}\|\cos{\theta}+\|\mathbf{d}\|\cos{\theta^\prime}=\mathbf{d}\cdot(\hat n-\hat n^\prime)=m\lambda</math>~~

	~~where <math>m\~~in~~\mathbb{Z}</math>~~. ~~Multiplying the above by <math>2\pi/\lambda</math> we formulate the condition in terms of the wave vectors <math>\mathbf{k}</math> and~~ <~~math~~>~~\mathbf{k^\prime}~~<~~/math~~>:

	:~~<math>\mathbf{d}\cdot(\mathbf{k}-\mathbf{k^\prime})=2\pi m</math>~~

	[~~[File~~:~~Bragg plane_illustration.png\|thumb\|300px\|The Bragg plane in blue, with its associated reciprocal lattice vector K.]]~~

	Now consider that a crystal is an array of scattering centres, each at a point in the [[Bravais lattice]]. We can set one of the scattering centres as the origin of an array. Since the lattice points are displaced by the Bravais lattice vectors <math>\mathbf{R}</~~math>, scattered waves interfere constructively when the above condition holds simultaneously for all values of <math>\mathbf{R}<~~/~~math> which are Bravais lattice vectors, the condition then becomes:~~

	~~:<math>\mathbf{R}\cdot(\mathbf{k}~~-~~\mathbf{k^\prime})=2\pi m</math>~~

	~~An equivalent statement (see [[Reciprocal_lattice#Mathematical_description\|mathematical description of the reciprocal lattice]]) is to say that:~~

	~~:<math>e^{i(\mathbf{k}~~-~~\mathbf{k^\prime})\cdot\mathbf{R}}=1</math>~~

	By comparing this equation with the definition of a reciprocal lattice vector, we see that constructive interference occurs if <math>\mathbf{K}=\mathbf{k}-\mathbf{k^\prime}</math> is a vector of the reciprocal lattice. ~~We notice that <math>\mathbf{k}<~~/~~math> and <math>\mathbf{k^\prime}<~~/math> have the same magnitude, we can restate the Von Laue formulation as requiring that the tip of incident wave vector <math>\mathbf{k}</math> must lie in the plane that is a perpendicular bisector of the reciprocal lattice vector <math>\mathbf{K}</math>. ~~This reciprocal space plane is the ''Bragg plane''.~~

	~~==References==~~
	~~{{reflist}}~~

	~~==See also==~~
	* [[X-ray crystallography]]
	* [[Reciprocal lattice]]
	* [[Bravais lattice]]
	* [[Powder diffraction]]
	* [[Kikuchi line]]
	* [[Brillouin zone]]


	~~[[Category:Crystallography]]~~
	~~[[Category:Geometry]]~~
	~~[[Category:Fourier analysis]]~~
	~~[[Category:Lattice points]]~~
	~~[[Category:Diffraction]~~]

en>Daddi Moussa Ider Abdallah: A video has been aded to illstrate the saddle node bifurcation

2013-11-17T08:41:04Z

A video has been aded to illstrate the saddle node bifurcation

← Older revision		Revision as of 10:41, 17 November 2013
Line 1:		Line 1:
	~~Nice to satisfy you~~, ~~I am Marvella Shryock~~. ~~California~~ is ~~our beginning location~~. ~~What I adore performing~~ is ~~to collect badges but I've been taking on new things lately~~. ~~For many years he~~'s ~~been operating~~ as a ~~receptionist~~.<br><br>~~My homepage~~ [~~http~~://~~www~~.~~gaysphere.net~~/~~blog~~/~~167593 gaysphere~~.~~net~~]		[[File:Vonlaue.png\|thumb\|300px\|Ray diagram of Von Laue formulation]]

			In [[physics]], a '''Bragg plane''' is a [[Plane (geometry)\|plane]] in [[reciprocal space]] which bisects one reciprocal lattice vector <math>\mathbf{K}</math>.<ref>{{Cite book
			\| last1 = Ashcroft \| first1 = Neil W.
			\| last2 = Mermin \| first2 = David
			\| title = Solid State Physics
			\| publisher = Brooks Cole
			\| edition = 1
			\| date = January 2, 1976
			\| pages = 96–100
			\| isbn = 0-03-083993-9}}</ref> It is relevant to define this plane as part of the definition of the Von Laue condition for [[Interference (wave propagation)\|diffraction peaks]] in [[X-ray_crystallography\|x-ray diffraction crystallography]].

			Considering the diagram at right, the arriving [[x-ray]] [[plane wave]] is defined by:

			:<math>e^{i\mathbf{k}\cdot\mathbf{r}}=\cos {(\mathbf{k}\cdot\mathbf{r})} +i\sin {(\mathbf{k}\cdot\mathbf{r})}</math>

			Where <math>\mathbf{k}</math> is the incident wave vector given by:

			:<math>\mathbf{k}=\frac{2\pi}{\lambda}\hat n</math>

			where <math>\lambda</math> is the [[wavelength]] of the incident [[photon]]. While the [[Bragg's law\|Bragg formulation]] assumes a unique choice of direct lattice planes and [[specular reflection]] of the incident X-rays, the Von Laue formula only assumes monochromatic light and that each scattering center acts as a source of secondary wavelets as described by the [[Huygens principle]]. Each scattered wave contributes to a new plane wave given by:

			:<math>\mathbf{k^\prime}=\frac{2\pi}{\lambda}\hat n^\prime</math>

			The condition for constructive interference in the <math>\hat n^\prime</math> direction is that the path difference between the photons is an integer multiple (m) of their wavelength. We know then that for constructive interference we have:

			:<math>\|\mathbf{d}\|\cos{\theta}+\|\mathbf{d}\|\cos{\theta^\prime}=\mathbf{d}\cdot(\hat n-\hat n^\prime)=m\lambda</math>

			where <math>m\in\mathbb{Z}</math>. Multiplying the above by <math>2\pi/\lambda</math> we formulate the condition in terms of the wave vectors <math>\mathbf{k}</math> and <math>\mathbf{k^\prime}</math>:

			:<math>\mathbf{d}\cdot(\mathbf{k}-\mathbf{k^\prime})=2\pi m</math>

			[[File:Bragg plane_illustration.png\|thumb\|300px\|The Bragg plane in blue, with its associated reciprocal lattice vector K.]]

			Now consider that a crystal is an array of scattering centres, each at a point in the [[Bravais lattice]]. We can set one of the scattering centres as the origin of an array. Since the lattice points are displaced by the Bravais lattice vectors <math>\mathbf{R}</math>, scattered waves interfere constructively when the above condition holds simultaneously for all values of <math>\mathbf{R}</math> which are Bravais lattice vectors, the condition then becomes:

			:<math>\mathbf{R}\cdot(\mathbf{k}-\mathbf{k^\prime})=2\pi m</math>

			An equivalent statement (see [[Reciprocal_lattice#Mathematical_description\|mathematical description of the reciprocal lattice]]) is to say that:

			:<math>e^{i(\mathbf{k}-\mathbf{k^\prime})\cdot\mathbf{R}}=1</math>

			By comparing this equation with the definition of a reciprocal lattice vector, we see that constructive interference occurs if <math>\mathbf{K}=\mathbf{k}-\mathbf{k^\prime}</math> is a vector of the reciprocal lattice. We notice that <math>\mathbf{k}</math> and <math>\mathbf{k^\prime}</math> have the same magnitude, we can restate the Von Laue formulation as requiring that the tip of incident wave vector <math>\mathbf{k}</math> must lie in the plane that is a perpendicular bisector of the reciprocal lattice vector <math>\mathbf{K}</math>. This reciprocal space plane is the ''Bragg plane''.

			==References==
			{{reflist}}

			==See also==
			* [[X-ray crystallography]]
			* [[Reciprocal lattice]]
			* [[Bravais lattice]]
			* [[Powder diffraction]]
			* [[Kikuchi line]]
			* [[Brillouin zone]]


			[[Category:Crystallography]]
			[[Category:Geometry]]
			[[Category:Fourier analysis]]
			[[Category:Lattice points]]
			[[Category:Diffraction]]

en>ZéroBot: r2.7.1) (robot Adding: ru:Седлоузловая бифуркация

2011-07-11T23:44:25Z

r2.7.1) (robot Adding: ru:Седлоузловая бифуркация

New page

Nice to satisfy you, I am Marvella Shryock. California is our beginning location. What I adore performing is to collect badges but I've been taking on new things lately. For many years he's been operating as a receptionist.<br><br>My homepage [http://www.gaysphere.net/blog/167593 gaysphere.net]