Arithmetic number - Revision history

en>David Eppstein: break up overlong paragraph, sectionize, give example, unstub, remove unsolved problem cat (no clear unsolved problem is stated here)

2014-12-06T23:59:19Z

break up overlong paragraph, sectionize, give example, unstub, remove unsolved problem cat (no clear unsolved problem is stated here)

← Older revision		Revision as of 01:59, 7 December 2014
Line 1:		Line 1:
	Cantilever magnetometry is the use of a [[cantilever]] to measure the [[magnetic moment]] of magnetic particles. The cantilevers used for this technique are about as long as a human hair is wide (100 [[µm]]) or smaller. On the end of cantilever is glued a small piece of [[ferromagnetic]] material (called a particle), ~~which interacts with external magnetic fields and creates torque on the cantilever. These torques cause the cantilever~~ to ~~oscillate faster or slower, depending~~ on the orientation of the particle's moment with respect to the external field, and the magnitude of the moment. The magnitude of the moment and [[magnetic anisotropy]] of the particle are then deduced by measuring the cantilever's oscillation frequency while varying the external field, and finding a best fit to the resultant data.<~~ref name="Rugar"~~>~~{{cite journal\|last=Rugar\|first=Dan\|coauthors=Stipe~~, ~~Mamin, Stowe, Kenny\|journal=Physical Review Letters\|year=2001\|volume=86\|issue=13\|url=http://phy-dhcp-35-17.mps.ohio-state.edu/HammelGroupWiki/images/6/6a/PhysRevLett.86.2874.pdf}}</ref>~~		In fact, more millionaires made their fortunes in real estate than in any other business according to statistics on the Internet.<br>PabloPicker, based on this: How rare is it for volatile stocks to actually sell at the correct price? Gain Hunter seems to be right about this one so far. They have followed from the beginning and have been accurately forecasting it.<br>http://www.shlgrouptr.com/warm/?p=275 <br /> http://www.shlgrouptr.com/warm/?p=6 <br /> http://www.shlgrouptr.com/warm/?p=235 <br /> http://www.shlgrouptr.com/warm/?p=142 <br /> http://www.shlgrouptr.com/warm/?p=355 <br /><br><br>Should you loved this article and you would want to receive more details about [http://www.bendtrapclub.com/cheap/ugg.asp Cheap Uggs Boots] i implore you to visit the page.
	[[File:Cantilever in external magnetic field.png\|thumb\|350px\|alt=Alt text\|A cantilever with a magnetic particle oscillating in an external magnetic field. Many setups do not have a modulation coil as pictured above, but some use capacitive coupling of a nearby wire instead of the modulation coil or piezoelectric transducer (PZT) to drive the cantilever]]
	~~To understand~~ this ~~technique, it is useful to think of a pendulum~~: ~~on earth it oscillates with one frequency, while if we put the same pendulum on, say, the moon, it would oscillate with a different,~~ ~~slower frequency. This~~ is ~~because the mass on~~ the ~~end of the pendulum interacts with the external gravitational field, much as the magnetic moment in this technique interacts with the external magnetic field. If one were~~ to ~~measure the frequency of a pendulum (and the length), they could extract the gravitational field since <math>\omega=\sqrt{g/l}\Rightarrow g=\omega^2 l</math>, whereas in~~ this ~~technique~~ one ~~measures the frequency and extracts the moment~~.

	~~==Cantilever Equation of Motion==~~
	As the ~~cantilever oscillates back~~ and ~~forth~~ it ~~arches into hyperbolic curves with the salient feature that a tangent to the end of the cantilever always intersects one point along the middle axis~~. ~~From this we define the effective length of the cantilever,~~ <~~math~~>~~L_{e}<~~/~~math>, to be the distance from this point to the end of the cantilever (see image to the right)~~. ~~The Lagrangian for this system is then given by~~
	~~{{NumBlk\|:\|<math>~~
	\mathcal{L}=T-V=\frac{1}{2}m\underbrace{(L_{e}\dot{\theta})^{2}}_{v^2}-\left(\underbrace{\frac{1}{2}k ( L_{e} \theta )^{2}}_{\begin{smallmatrix}\text{Restoring}\\ \text{Energy}\end{smallmatrix}}-\underbrace{\mu B \cos{(\theta-\beta)}}_{\begin{smallmatrix}\text{Zeeman}\\ \text{Energy}\end{smallmatrix}}+\underbrace{K_uV\sin^2{\beta}}_{\begin{smallmatrix}\text{Anisotropy}\\ \text{Energy}\end{smallmatrix}}\right)
	</~~math>\|Eq. 1}}~~
	~~where <math>m<~~/~~math> is the effective cantilever mass, <math>V~~</~~math~~> ~~is the volume of the particle, <math>k<~~/~~math> is the cantilever-constant and <math>\mu<~~/~~math> is the magnetic moment of the particle~~. ~~To find the equation of motion we note that we have two variables, <math>\theta<~~/~~math> and <math>\beta<~~/~~math> so there are two corresponding Lagrangian equations which need to be solved as a system of equations,~~
	~~:<math>~~
	~~\frac{d}{d t}\frac{\partial \mathcal{L}}{\partial \dot{\beta}}~~=~~\frac{\partial \mathcal{L}}{\partial \beta}\Rightarrow 0=-\mu B \sin{ (\theta-\beta)}+2K_uV\underbrace{\sin\beta \cos\beta}_{\approx \beta} \Rightarrow~~
	</~~math~~>
	~~{{NumBlk\|~~:~~\|<math>\beta=\frac{B\theta}{B+H_k}<~~/~~math>\|Eq. 2}}~~

	~~where we have defined <math>H_k\equiv\frac{2K_uV}{\mu}<~~/~~math>~~.

	~~We can plug Eq~~. ~~2 into our Lagrangian, which then becomes a function of <math>\theta<~~/~~math> only. Then <math>(\theta - \beta)~~=~~\frac{\theta H_k}{B+H_k}~~</~~math~~>~~, and we have~~
	:~~<math>~~
	~~\frac{d}{d t}\frac{\partial \mathcal{L}}{\partial \dot{\theta}}=\frac{\partial \mathcal{L}}{\partial \theta}\Rightarrow~~
	</~~math>~~
	~~:<math>~~
	mL_{e}^{2}\ddot{\theta}=-kL_{e}^{2}\theta-\mu B \sin{\left(\frac{\theta H_k}{B+H_k}\right)}\left(\frac{H_k}{B+H_k}\right)-2K_uV\sin{\left(\frac{\theta B}{B+H_k}\right)}\cos{\left(\frac{\theta B}{B+H_k}\right)}\left(\frac{B}{B+H_k}\right)\Rightarrow
	</~~math>~~
	~~:<math>\approx -kL_{e}^{2}\theta-\mu B \theta\left(\frac{ H_k}{B+H_k}\right)^2-\mu H_k\theta\left(\frac{B}{B+H_k}\right)^2\Rightarrow<~~/~~math>~~
	~~<math>~~
	~~\ddot{\theta}+\theta\left( \frac{kL_{e}^{2}+\frac{\mu BH_k}{(B+H_k)}}{mL_{e}^{2}}\right)=\ddot{\theta}+\theta\left( \omega_o^2 +\frac{\mu BH_k}{mL_{e}^{2}(B+H_k)}\right)~~=0,
	</~~math~~>

	or
	~~{{NumBlk\|~~:~~\| <math>\ddot{\theta}+\omega^2\theta=0,<~~/~~math>\|Eq~~. ~~3}}~~
	~~where <math>\omega^{2}\equiv\left( \omega_o^2 +\frac{\mu BH_k}{mL_{e}^{2}(B+H_k)}\right)=\omega_o^2\left( 1 +\frac{\mu BH_k}{kL_{e}^{2}(B+H_k)}\right)<~~/~~math>. The solution to this differential equation is <math>\theta(t)~~=~~c_{1}\cos\omega t+c_{2}\sin\omega t~~</~~math~~> ~~where~~ <~~math~~>~~c_{1}~~<~~/math~~> ~~and <math>c_{2}</math> are coefficients determined by the initial conditions. It is interesting to note that the motion of a simple pendulum is similarly described by~~ this ~~differential equation~~ and ~~solution in the small angle approximation.~~

	~~We can use the binomial expansion~~ to ~~rewrite <math>\omega<~~/~~math>,~~
	~~:<math>\omega=\omega_o\left( 1 +\frac{\mu BH_k}{kL_{e}^{2}(B+H_k)}\right)^{1~~/~~2}\approx \omega_o\left( 1 +\frac{\mu BH_k}{2kL_{e}^{2}(B+H_k)}+~~..~~.\right)\Rightarrow<~~/~~math>~~
	~~{{Equation box 1~~
	~~\|equation={{NumBlk\|:\|<math>\frac{\omega-\omega_o}{\omega_o}=\frac{\Delta \omega}{\omega_o}=\frac{\mu HH_k}{2kL_e^2(H+H_k)},<~~/~~math>\|Eq~~. ~~4}}}}~~

	~~which is~~ the ~~form as seen in literature, for example equation 2 in the paper "Magnetic Dissipation and Fluctuations in Individual Nanomagnets Measured~~
	~~by Ultrasensitive Cantilever Magnetometry" <ref name="Rugar" />~~

	~~{{Reflist}}~~



	~~[[Category:Magnetometers]]~~

en>Toshio Yamaguchi: /* References */ adding natural number classes navbox

2013-05-30T20:20:23Z

References: adding natural number classes navbox

← Older revision		Revision as of 22:20, 30 May 2013
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It is ~~thе place~~ the ~~stain уou may chat with eveгy single 1~~ in the ~~men and women nowadays inside of of yoսr extremely personal consciousness acquiring а sexual attraction οf mеn and women ϲurrently ԝith each othеr whіlе սsing relatеd sexual~~. ~~Ҭhis facility pгovides chat ɑt a гelated time as courting professional expertise ɑll through~~ the ~~wɦole entire world~~. ~~Іt actuɑlly~~ is ~~a sort of neighbourhood thе ɑrea you mіght locate ɑ wҺole wholе lot of mеn~~ and ~~women of one pаrticular's respective гeally individual focus and ԝay extra things linked on your οwn individual keen fascination.~~		Cantilever magnetometry is the use of a [[cantilever]] to measure the [[magnetic moment]] of magnetic particles. The cantilevers used for this technique are about as long as a human hair is wide (100 [[µm]]) or smaller. On the end of cantilever is glued a small piece of [[ferromagnetic]] material (called a particle), which interacts with external magnetic fields and creates torque on the cantilever. These torques cause the cantilever to oscillate faster or slower, depending on the orientation of the particle's moment with respect to the external field, and the magnitude of the moment. The magnitude of the moment and [[magnetic anisotropy]] of the particle are then deduced by measuring the cantilever's oscillation frequency while varying the external field, and finding a best fit to the resultant data.<ref name="Rugar">{{cite journal\|last=Rugar\|first=Dan\|coauthors=Stipe, Mamin, Stowe, Kenny\|journal=Physical Review Letters\|year=2001\|volume=86\|issue=13\|url=http://phy-dhcp-35-17.mps.ohio-state.edu/HammelGroupWiki/images/6/6a/PhysRevLett.86.2874.pdf}}</ref>
			[[File:Cantilever in external magnetic field.png\|thumb\|350px\|alt=Alt text\|A cantilever with a magnetic particle oscillating in an external magnetic field. Many setups do not have a modulation coil as pictured above, but some use capacitive coupling of a nearby wire instead of the modulation coil or piezoelectric transducer (PZT) to drive the cantilever]]
			To understand this technique, it is useful to think of a pendulum: on earth it oscillates with one frequency, while if we put the same pendulum on, say, the moon, it would oscillate with a different, slower frequency. This is because the mass on the end of the pendulum interacts with the external gravitational field, much as the magnetic moment in this technique interacts with the external magnetic field. If one were to measure the frequency of a pendulum (and the length), they could extract the gravitational field since <math>\omega=\sqrt{g/l}\Rightarrow g=\omega^2 l</math>, whereas in this technique one measures the frequency and extracts the moment.

			==Cantilever Equation of Motion==
			As the cantilever oscillates back and forth it arches into hyperbolic curves with the salient feature that a tangent to the end of the cantilever always intersects one point along the middle axis. From this we define the effective length of the cantilever, <math>L_{e}</math>, to be the distance from this point to the end of the cantilever (see image to the right). The Lagrangian for this system is then given by
			{{NumBlk\|:\|<math>
			\mathcal{L}=T-V=\frac{1}{2}m\underbrace{(L_{e}\dot{\theta})^{2}}_{v^2}-\left(\underbrace{\frac{1}{2}k ( L_{e} \theta )^{2}}_{\begin{smallmatrix}\text{Restoring}\\ \text{Energy}\end{smallmatrix}}-\underbrace{\mu B \cos{(\theta-\beta)}}_{\begin{smallmatrix}\text{Zeeman}\\ \text{Energy}\end{smallmatrix}}+\underbrace{K_uV\sin^2{\beta}}_{\begin{smallmatrix}\text{Anisotropy}\\ \text{Energy}\end{smallmatrix}}\right)
			</math>\|Eq. 1}}
			where <math>m</math> is the effective cantilever mass, <math>V</math> is the volume of the particle, <math>k</math> is the cantilever-constant and <math>\mu</math> is the magnetic moment of the particle. To find the equation of motion we note that we have two variables, <math>\theta</math> and <math>\beta</math> so there are two corresponding Lagrangian equations which need to be solved as a system of equations,
			:<math>
			\frac{d}{d t}\frac{\partial \mathcal{L}}{\partial \dot{\beta}}=\frac{\partial \mathcal{L}}{\partial \beta}\Rightarrow 0=-\mu B \sin{ (\theta-\beta)}+2K_uV\underbrace{\sin\beta \cos\beta}_{\approx \beta} \Rightarrow
			</math>
			{{NumBlk\|:\|<math>\beta=\frac{B\theta}{B+H_k}</math>\|Eq. 2}}

			where we have defined <math>H_k\equiv\frac{2K_uV}{\mu}</math>.

			We can plug Eq. 2 into our Lagrangian, which then becomes a function of <math>\theta</math> only. Then <math>(\theta - \beta)=\frac{\theta H_k}{B+H_k}</math>, and we have
			:<math>
			\frac{d}{d t}\frac{\partial \mathcal{L}}{\partial \dot{\theta}}=\frac{\partial \mathcal{L}}{\partial \theta}\Rightarrow
			</math>
			:<math>
			mL_{e}^{2}\ddot{\theta}=-kL_{e}^{2}\theta-\mu B \sin{\left(\frac{\theta H_k}{B+H_k}\right)}\left(\frac{H_k}{B+H_k}\right)-2K_uV\sin{\left(\frac{\theta B}{B+H_k}\right)}\cos{\left(\frac{\theta B}{B+H_k}\right)}\left(\frac{B}{B+H_k}\right)\Rightarrow
			</math>
			:<math>\approx -kL_{e}^{2}\theta-\mu B \theta\left(\frac{ H_k}{B+H_k}\right)^2-\mu H_k\theta\left(\frac{B}{B+H_k}\right)^2\Rightarrow</math>
			<math>
			\ddot{\theta}+\theta\left( \frac{kL_{e}^{2}+\frac{\mu BH_k}{(B+H_k)}}{mL_{e}^{2}}\right)=\ddot{\theta}+\theta\left( \omega_o^2 +\frac{\mu BH_k}{mL_{e}^{2}(B+H_k)}\right)=0,
			</math>

			or
			{{NumBlk\|:\| <math>\ddot{\theta}+\omega^2\theta=0,</math>\|Eq. 3}}
			where <math>\omega^{2}\equiv\left( \omega_o^2 +\frac{\mu BH_k}{mL_{e}^{2}(B+H_k)}\right)=\omega_o^2\left( 1 +\frac{\mu BH_k}{kL_{e}^{2}(B+H_k)}\right)</math>. The solution to this differential equation is <math>\theta(t)=c_{1}\cos\omega t+c_{2}\sin\omega t</math> where <math>c_{1}</math> and <math>c_{2}</math> are coefficients determined by the initial conditions. It is interesting to note that the motion of a simple pendulum is similarly described by this differential equation and solution in the small angle approximation.

			We can use the binomial expansion to rewrite <math>\omega</math>,
			:<math>\omega=\omega_o\left( 1 +\frac{\mu BH_k}{kL_{e}^{2}(B+H_k)}\right)^{1/2}\approx \omega_o\left( 1 +\frac{\mu BH_k}{2kL_{e}^{2}(B+H_k)}+...\right)\Rightarrow</math>
			{{Equation box 1
			\|equation={{NumBlk\|:\|<math>\frac{\omega-\omega_o}{\omega_o}=\frac{\Delta \omega}{\omega_o}=\frac{\mu HH_k}{2kL_e^2(H+H_k)},</math>\|Eq. 4}}}}

			which is the form as seen in literature, for example equation 2 in the paper "Magnetic Dissipation and Fluctuations in Individual Nanomagnets Measured
			by Ultrasensitive Cantilever Magnetometry" <ref name="Rugar" />

			{{Reflist}}



			[[Category:Magnetometers]]

Deltahedron: first few numbers, cite OEIS

2012-08-25T16:33:45Z

first few numbers, cite OEIS

New page

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