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| 1 | +\chapter{Theory} |
| 2 | +\label{chap:theory} |
| 3 | + |
| 4 | +\section{Magnetic levitation} |
| 5 | +\label{sec:magnetic_levitation} |
| 6 | +Obtaining and then maintaining stable magnetic levitation requires three fields: a static homogenous field to align the magnet ($\vec{B_0}$); an alternating field containing the saddle point ($\vec{B_1}$); and a gradient field to counter the gravitational offset ($\vec{B_2}$). This follows from simulations and our previous experience \cite{perdriat,mart,eli}. Combining this with the magnetic moment of the levitated particle $\vec{\mu}$ allows us to derive (up to second order) the magnetic potential. Averaging over the oscillation period gives allows us to derive the associated eigenfrequencies. |
| 7 | + |
| 8 | +\subsection{Fields} |
| 9 | +\label{subsec:fields} |
| 10 | +The homogeneous field is given in \autoref{eq:homogeneous_field}. It is oriented in the $z$-axis with magnitude $B_0$. Homogenous fields can (locally) be created using Helmholtz coils. |
| 11 | +\begin{equation} |
| 12 | + \vec{B_0}(\vec{r}, t) = \vec{B_0} = B_0 \zhat |
| 13 | + \label{eq:homogeneous_field} |
| 14 | +\end{equation} |
| 15 | + |
| 16 | +Rotating a saddle point sufficiently fast effectively creates a (local) minimum. The field is given in \autoref{eq:saddle_point}. $B_1''$ is the curvature of $\vec{B_1}$ and $\Omega / 2\pi$ the frequency of the oscillation. The field can be created using two loops of wire with a current in opposite directions in the same plane, a so called magnetic Paul Trap. These loops are the main focus of this thesis. Given the radius of the inner loop $r_1$ we can express the curvature as $B_1'' = -\frac{9}{16}\mu_0i_1/r_1^3$ where $i_1$ is the current through the loop\footnote{This uses the requirement that $i_1/i_2 = -r_1/r_2$ and that $i_1$ and $i_2$ oppose each other (clockwise and anti-clockwise).}. |
| 17 | +\begin{equation} |
| 18 | + \vec{B_1}(\vec{r}, t) = \frac{B_1''}{2} \begin{pmatrix} |
| 19 | + -xz \\ |
| 20 | + +yz \\ |
| 21 | + z^2 - \frac{1}{2}\left(x^2 + y^2\right) |
| 22 | + \end{pmatrix} \cos(\Omega t) |
| 23 | + \label{eq:saddle_point} |
| 24 | +\end{equation} |
| 25 | + |
| 26 | +Finally we have the gradient field, given in \autoref{eq:gradient_field}. The required gradient can be expressed as $B_2' = mg/\mu$ where $m$ is the mass of the levitated particle, $g$ the gravitational acceleration and $\mu$ the magnetic moment of the particle. The gradient serves to offset the effect of gravity. As such the gradient is oriented in the $z$-axis. The gradient field can be created by using a larger current in one of the Helmholtz coils. |
| 27 | +\begin{equation} |
| 28 | + \vec{B_2}(\vec{r}, t) = B_2' \begin{pmatrix} |
| 29 | + -x / 2 \\ |
| 30 | + -y / 2 \\ |
| 31 | + z |
| 32 | + \end{pmatrix} |
| 33 | + \label{eq:gradient_field} |
| 34 | +\end{equation} |
| 35 | + |
| 36 | +\subsection{Magnetic potential} |
| 37 | +\label{subsec:magnetic_moment} |
| 38 | +Using the $zyz$ convention for the Euler angles ($\alpha$, $\beta$, $\gamma$) we can express the magnetic moment as in \autoref{eq:magnetic_moment}. In this equation $\tilde\beta = \beta - \pi/2$. |
| 39 | +\begin{equation} |
| 40 | + \vec{\mu} = -\mu \begin{pmatrix} |
| 41 | + -\cos(\alpha)\sin(\tilde\beta)\cos(\gamma) - \sin(\alpha)\sin(\gamma) \\ |
| 42 | + \cos(\alpha)\sin(\gamma) - \sin(\alpha)\sin(\tilde\beta)\cos(\gamma) \\ |
| 43 | + -\cos(\tilde\beta)\cos(\gamma) |
| 44 | + \end{pmatrix} |
| 45 | + \label{eq:magnetic_moment} |
| 46 | +\end{equation} |
| 47 | + |
| 48 | +By taking the inner product of the magnetic moment and the fields we can derive the magnetic potential. This is given in \autoref{eq:magnetic_potential}. The potential is a function of the position of the particle $\vec{r}$ and the orientation of the magnetic moment $\vec{\mu}$. |
| 49 | +\begin{equation} |
| 50 | + E_\text{mag}(\vec{r}, \vec{\mu}) = \mu B_0 \left(\frac{\gamma^2}{2} + \frac{\tilde\beta^2}{2}\right) - \frac{\mu B_1''}{2} \left(z^2 - \frac{1}{2}\left(x^2 + y^2\right)\right)\cos(\Omega t) |
| 51 | + \label{eq:magnetic_potential} |
| 52 | +\end{equation} |
| 53 | + |
| 54 | +\subsection{Eigenfrequencies} |
| 55 | +\label{subsec:eigenfrequencies} |
| 56 | +Starting from \autoref{eq:magnetic_potential} and averaging over the oscillation period we can derive the eigenfrequencies. They are given in \autoref{eq:eigenfrequencies}. In these equations $a$ is the radius of the levitated particle (such that $V \sim a^3$) and $\rho_m$ is the density of the particle. The eigenfrequencies associated with the orientation of the magnet depend on $B_0$, which intuitively makes sense since the orientation of the magnet is determined by the homogenous field. The eigenfrequencies associated with the position of the magnet depend on $B_1''$, which also makes sense since the alternating field is what restricts the movement of the particle. |
| 57 | +\begin{equation} |
| 58 | + \begin{gathered} |
| 59 | + \omega_\gamma = \omega_{\tilde\beta} = \sqrt{\frac{5}{2}\frac{B_0B_\text{sat}}{\mu_0 \rho_m a^2}} \\ |
| 60 | + \omega_z = 2\omega_x = 2\omega_y = \frac{\Omega \abs{q_z}}{2\sqrt{2}} = \frac{1}{\sqrt{2}}\frac{B_1''B_\text{sat}}{\mu_0\rho_m\Omega} |
| 61 | + \end{gathered} |
| 62 | + \label{eq:eigenfrequencies} |
| 63 | +\end{equation} |
| 64 | + |
| 65 | +The time average is valid if $\abs{q_z}$, as defined in equation \ref{eq:q-factors}, is less than or equal to $0.4$. |
| 66 | +\begin{equation} |
| 67 | + q_z = -2q_x = -2q_y = \frac{2}{\Omega^2}\frac{B_1''B_{\text{sat}}}{\mu_0\rho_m} |
| 68 | + \label{eq:q-factors} |
| 69 | +\end{equation} |
| 70 | + |
| 71 | +TODO: $\omega_\alpha$? |
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