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翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Cantilever Magnetometry ： ウィキペディア英語版 Cantilever Magnetometry Cantilever magnetometry is the use of a cantilever to measure the magnetic moment of magnetic particles. On the end of cantilever is attached a small piece of magnetic material, which interacts with external magnetic fields and exherts 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 material can be deduced by measuring the cantilever's oscillation frequency versus external field. A useful, although limited analogy is that of a pendulum: on earth it oscillates with one frequency, while the same pendulum on, say, the moon, would oscillate with a slower frequency. This is because the mass on the end of the pendulum interacts with the external gravitational field, much as a magnetic moment interacts with an external magnetic field. ==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, $L\_$, 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 _-\left(\underbracek ( L_ \theta )^}_\\ \text\end}-\underbrace\text\\ \text\end}+\underbrace\text\\ \text\end}\right) |Eq. 1}} where $m$ is the effective cantilever mass, $V$ is the volume of the particle, $k$ is the cantilever-constant and $\backslash mu$ is the magnetic moment of the particle. To find the equation of motion we note that we have two variables, $\backslash theta$ and $\backslash beta$ so there are two corresponding Lagrangian equations which need to be solved as a system of equations, :$$ \frac\frac}=\frac\Rightarrow 0=-\mu B \sin+2K_uV\underbrace_ \Rightarrow where we have defined $H\_k\backslash equiv\backslash frac$. We can plug Eq. 2 into our Lagrangian, which then becomes a function of $\backslash theta$ only. Then $(\backslash theta\; -\; \backslash beta)=\backslash frac$, and we have :$$ \frac\frac}=\frac\Rightarrow :$$ mL_^\ddot=-kL_^\theta-\mu B \sin\right)}\left(\frac\right)-2K_uV\sin\right)}\cos\right)}\left(\frac\right)\Rightarrow :$\backslash approx\; -kL\_^\backslash theta-\backslash mu\; B\; \backslash theta\backslash left(\backslash frac\backslash right)^2-\backslash mu\; H\_k\backslash theta\backslash left(\backslash frac\backslash right)^2\backslash Rightarrow$ $$ \ddot+\theta\left( \frac+\frac}}\right)=\ddot+\theta\left( \omega_o^2 +\frac(B+H_k)}\right)=0, or where $\backslash omega^\backslash equiv\backslash left(\; \backslash omega\_o^2\; +\backslash frac(B+H\_k)\}\backslash right)=\backslash omega\_o^2\backslash left(\; 1\; +\backslash frac(B+H\_k)\}\backslash right)$. The solution to this differential equation is $\backslash theta(t)=c\_\backslash cos\backslash omega\; t+c\_\backslash sin\backslash omega\; t$ where $c\_$ and $c\_$ 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 $\backslash omega$, :$\backslash omega=\backslash omega\_o\backslash left(\; 1\; +\backslash frac(B+H\_k)\}\backslash right)^\backslash approx\; \backslash omega\_o\backslash left(\; 1\; +\backslash frac(B+H\_k)\}+...\backslash right)\backslash Rightarrow$ 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" 〔 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Cantilever Magnetometry」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.