Many small fixes.

Author: jvdoorn

chapters/results.tex

@@ -2,21 +2,21 @@ \chapter{Results}
 \label{chap:results}
 The datasheet of our \ce{NdFeB} particles specifies that their residual magnetic field is \qtyrange{730}{760}{\milli\tesla}. Full saturation (\qty{>95}{\percent}) is achieved when a \qty{2}{\tesla} field is applied\cite{magnequench}. Our method of magnetising the particle was limited to a \qty{1.4}{\tesla} field. As such, it is likely that we did not achieve the full saturation of the particle and that the residual magnetic field is lower than specified. Assuming a roughly linear relation between the two values, we estimate the residual magnetic field of our particles to be \qtyrange{510}{530}{\milli\tesla}. The density of the particles is given as \qty{7430}{\kilo\gram\per\cubic\meter}\cite{magnequench}.
 
-These values, together with the simulation of the magnetic field distribution in the trap, allow us to estimate the required driving frequency and the resulting eigen frequency. \autoref{fig:magnetic-field-curvature} shows the $z$-component of the ac magnetic field ($\vec{B_1}$) for a range of $\chi$ values as well as the curvature. It is important to note that the \xmode and \ymode have a negative curvature, whilst the \zmode has a positive curvature. This means that $x$ and $y$ are not stable. Half a period later however this will have changed, and the particle will be stable in the $x$- and $y$-direction instead of the $z$-direction. This is what leads to the ponderomotive effect. The curves for $y$ are not symmetric because the slit is in the direction of the \ymode.
+These values, together with the simulation of the magnetic field distribution in the trap, allow us to estimate the required driving frequency and the resulting eigenfrequency. \autoref{fig:magnetic-field-curvature} shows the $z$-component of the ac magnetic field ($\vec{B_1}$) for a range of $\xi$ values as well as the curvature. It is important to note that the \xmode and \ymode have a negative curvature, whilst the \zmode has a positive curvature. This means that $x$ and $y$ are not stable. Half a period later however this will have changed, and the particle will be stable in the $x$- and $y$-direction instead of the $z$-direction. This is what leads to the ponderomotive effect. The curves for $y$ are not symmetric because the slit is in the direction of the \ymode.
 
 Using \autoref{eq:eigenfrequencies} and \autoref{eq:frequency-constraint} we can then estimate the required driving frequency and the resulting eigen frequencies. These results are shown in \autoref{fig:eigen-frequency-xi-dependence}.
 
 \begin{figure}
     \centering
     \includegraphics{figures/data/magnetic_field_curvature.pdf}
-    \caption{the \textbf{top} figures from left to right show $z$-component of the magnetic field evaluated along a line parallel to the $x$-, $y$- and $z$-axis respectively through the origin. The gray zones indicate the boundaryies of the trap. The \textbf{bottom left} figure shows the curvature of the magnetic field for these calculations evaluated in the extrema. The simulations were performed for $i_1=\qty{0.5}{\ampere}$ and $\chi$ between \numrange{0.5}{5}. The curves have been labelled with the corresponding value for $\chi$. In addition to this the offset in $y$ due to the trap asymmetry is shown in the \textbf{bottom right} figure.}
+    \caption{the \textbf{top} figures from left to right show $z$-component of the magnetic field evaluated along a line parallel to the $x$-, $y$- and $z$-axis respectively through the origin. The gray zones indicate the boundaryies of the trap. The \textbf{bottom left} figure shows the curvature of the magnetic field for these calculations evaluated in the extrema. The simulations were performed for $i_1=\qty{0.5}{\ampere}$ and $\xi$ between \numrange{0.5}{5}. The curves have been labelled with the corresponding value for $\xi$. In addition to this the offset in $y$ due to the trap asymmetry is shown in the \textbf{bottom right} figure.}
     \label{fig:magnetic-field-curvature}
 \end{figure}
 
 \begin{figure}
     \centering
     \includegraphics{figures/data/eigen_frequency_xi_dependence.pdf}
-    \caption{Simulation results of the dependence of the eigen frequency on $\chi$. These results are based on the data presented in \autoref{fig:magnetic-field-curvature}. The \textbf{top} figure shows the minimal trapping frequency such that $q_z \leq 0.4$. In the \textbf{middle} the corresponding eigenfrequencies of the modes are shown for $\Omega / 2\pi = \qty{2.5}{\kilo\hertz}$. The \textbf{bottom} figure shows the deviation from the center of the \ymode. The dashed lines are a linear interpolation of the data and only serve as a guide to the eyes.}
+    \caption{Simulation results of the dependence of the eigenfrequency on $\xi$. These results are based on the data presented in \autoref{fig:magnetic-field-curvature}. The \textbf{top} figure shows the minimal trapping frequency such that $q_z \leq 0.4$. In the \textbf{middle} the corresponding eigenfrequencies of the modes are shown for $\Omega / 2\pi = \qty{2.5}{\kilo\hertz}$. The \textbf{bottom} figure shows the deviation from the center of the \ymode. The dashed lines are a linear interpolation of the data and only serve as a guide to the eyes.}
     \label{fig:eigen-frequency-xi-dependence}
 \end{figure}