Cantilever magnetometry

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, ${\displaystyle L_{e}}$, 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

${\displaystyle {\mathcal {L}}=T-V={\frac {1}{2}}m\underbrace {(L_{e}{\dot {\theta }})^{2}} _{v^{2}}-\left(\underbrace {{\frac {1}{2}}k(L_{e}\theta )^{2}} _{\begin{smallmatrix}{\text{Restoring}}\\{\text{Energy}}\end{smallmatrix}}-\underbrace {\mu B\cos {(\theta -\beta )}} _{\begin{smallmatrix}{\text{Zeeman}}\\{\text{Energy}}\end{smallmatrix}}+\underbrace {K_{u}V\sin ^{2}{\beta }} _{\begin{smallmatrix}{\text{Anisotropy}}\\{\text{Energy}}\end{smallmatrix}}\right)}$

(Eq. 1)

where ${\displaystyle m}$ is the effective cantilever mass, ${\displaystyle V}$ is the volume of the particle, ${\displaystyle k}$ is the cantilever-constant and ${\displaystyle \mu }$ is the magnetic moment of the particle. To find the equation of motion we note that we have two variables, ${\displaystyle \theta }$ and ${\displaystyle \beta }$ so there are two corresponding Lagrangian equations which need to be solved as a system of equations,

${\displaystyle {\frac {d}{dt}}{\frac {\partial {\mathcal {L}}}{\partial {\dot {\beta }}}}={\frac {\partial {\mathcal {L}}}{\partial \beta }}\Rightarrow 0=\mu B\sin {(\theta -\beta )}-2K_{u}V\underbrace {\sin \beta \cos \beta } _{\approx \beta }\Rightarrow }$

${\displaystyle \beta ={\frac {B\theta }{B+H_{k}}}}$

(Eq. 2)

where we have defined ${\displaystyle H_{k}\equiv {\frac {2K_{u}V}{\mu }}}$.

We can plug Eq. 1 into our Lagrangian, which then becomes a function of ${\displaystyle \theta }$ only. Then ${\displaystyle (\theta -\beta )={\frac {\theta H_{k}}{B+H_{k}}}}$, and we have

${\displaystyle {\frac {d}{dt}}{\frac {\partial {\mathcal {L}}}{\partial {\dot {\theta }}}}={\frac {\partial {\mathcal {L}}}{\partial \theta }}\Rightarrow }$

${\displaystyle mL_{e}^{2}{\ddot {\theta }}=-kL_{e}^{2}\theta -\mu B\sin {\left({\frac {\theta H_{k}}{B+H_{k}}}\right)}\left({\frac {H_{k}}{B+H_{k}}}\right)-2K_{u}V\sin {\left({\frac {\theta B}{B+H_{k}}}\right)}\cos {\left({\frac {\theta B}{B+H_{k}}}\right)}\left({\frac {B}{B+H_{k}}}\right)\Rightarrow }$

${\displaystyle \approx -kL_{e}^{2}\theta -\mu B\theta \left({\frac {H_{k}}{B+H_{k}}}\right)^{2}-\mu H_{k}\theta \left({\frac {B}{B+H_{k}}}\right)^{2}\Rightarrow }$

${\displaystyle {\ddot {\theta }}+\theta \left({\frac {kL_{e}^{2}+{\frac {\mu BH_{k}}{(B+H_{k})}}}{mL_{e}^{2}}}\right)={\ddot {\theta }}+\theta \left(\omega _{o}^{2}+{\frac {\mu BH_{k}}{mL_{e}^{2}(B+H_{k})}}\right)=0,}$

or

${\displaystyle {\ddot {\theta }}+\omega ^{2}\theta =0,}$

(Eq. 3)

where ${\displaystyle \omega ^{2}\equiv \left(\omega _{o}^{2}+{\frac {\mu BH_{k}}{mL_{e}^{2}(B+H_{k})}}\right)=\omega _{o}^{2}\left(1+{\frac {\mu BH_{k}}{kL_{e}^{2}(B+H_{k})}}\right)}$. The solution to this differential equation is ${\displaystyle \theta (t)=c_{1}\cos \omega t+c_{2}\sin \omega t}$ where ${\displaystyle c_{1}}$ and ${\displaystyle c_{2}}$ are coefficients determined by the initial conditions. 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 ${\displaystyle \omega }$,

${\displaystyle \omega =\omega _{o}\left(1+{\frac {\mu BH_{k}}{kL_{e}^{2}(B+H_{k})}}\right)^{1/2}\approx \omega _{o}\left(1+{\frac {\mu BH_{k}}{2kL_{e}^{2}(B+H_{k})}}+...\right)\Rightarrow }$

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".[1]