89 lines
4.1 KiB
TeX
89 lines
4.1 KiB
TeX
\documentclass[../main.tex]{subfiles}
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\graphicspath{{\subfix{../images/}}}
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\begin{document}
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\section{Results and Discussion}
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% Single particle
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We simulated the movement of particles confined in a Penning trap. All simulations used the initial conditions for particle 1 and 2 given in table \ref{tab:initial_condition_particles}.
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First we simulated a single particle for $50 \mu s$, approximating the particle's motion using the RK4 method. In addition we compared the motion of particle 1 with the analytical solution in figure \ref{fig:single_particle}. What we see is a complete overlap of the analytical solution completely overlap the approximated, suggest that the simulation result is good.
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% Add something about why the simulated result is good, cos(wt) when w is
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%
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\begin{table}[H]
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\centering
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\begin{tabular}[c]{lll}
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Particle & Position & Velocity \\
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\hline
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$p_{1}$ & $(20, 0, 20) \mu m$ & $(0, 25, 0) \mu m/ \mu s$ \\
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$p_{2}$ & $(25, 25, 0) \mu m$ & $(0, 40, 5) \mu m/ \mu s$ \\
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\hline
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\end{tabular}
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\caption{Initial position and velocity of particle 1 ($p_{1}$) particle 2 ($p_{2}$), where the analytical solution is given by $z(t) = z_{0} \cos (\omega_{z} t)$}
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\label{tab:initial_condition_particles}
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\end{table}
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\begin{figure}[H]
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\centering
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\includegraphics[width=\linewidth]{images/single_particle.pdf}
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\caption{A single particle in the Penning trap, approximated and analytical motion in z-direction.}
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\label{fig:single_particle}
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\end{figure}
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% Add equations of motion for particle with interaction eq. (18, 19, 20)
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% Multiple particles
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% Add initial condition of Penning trap
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We will now consider the Penning trap with initial conditions given in table \ref{tab:initial_condition_penning}, and simulate using one or two particles. In addition, we simulate two particles both with and without interactions, the result is found in figure \ref{fig:two_particles}. When we add interaction between the particles, they both still follow the same inherent path. However, we observe a small shift in both particle's movement.
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\begin{table}[H]
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\centering
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\begin{tabular}{lll}
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$B_0$ & $V_{0}$ & $d$ \\
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\hline
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\end{tabular}
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\caption{Caption}
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\label{tab:initial_condition_penning}
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\end{table}
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%
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\begin{figure}
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\centering
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\includegraphics[width=\linewidth]{images/plot_2_particles_xy.pdf}
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\caption{Movement of two particles in the xy-plane. $\hat{p}_{1}$ and $\hat{p}_{2}$ include particle interaction, whereas $p_{1}$ and $p_{2}$ does not include particle interaction.}
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\label{fig:two_particles}
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\end{figure}
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% Phase space plot
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When we simulate two particles, we can see the effect of interaction between the particles in the xy-plane in fig. \ref{fig:phase_space_2x} and in the z-direction in fig. \ref{fig:phase_space_2z}. What we observe is a very small shift in position for particle 1 in x-direction, whereas particle 2 does not have a visible shift. In the z-direction, however, the oscillation of particle 2 experience a greater shift. Particle 2 experience the force of particle 1 such that particle 2 moves larger distance.
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%
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\begin{figure}
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\centering
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\includegraphics[width=\linewidth]{images/phase_space_2_particles_x.pdf}
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\caption{Phase space plot of two particles in x-direction.}
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\label{fig:phase_space_2x}
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\end{figure}
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%
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\begin{figure}
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\centering
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\includegraphics[width=\linewidth]{images/phase_space_2_particles_z.pdf}
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\caption{Phase space plot of two particles in z-direction.}
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\label{fig:phase_space_2z}
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\end{figure}
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\begin{figure}
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\centering
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\includegraphics[width=\linewidth]{images/3d_plot.pdf}
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\caption{3D plot of particles-}
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\label{fig:3d_particles}
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\end{figure}
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\begin{figure}
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\centering
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\includegraphics[width=\linewidth]{images/phase_space_2_particles.pdf}
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\caption{Phase space plot of two particles.}
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\label{fig:phase_space_2}
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\end{figure}
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\begin{figure}
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\centering
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\includegraphics[width=\linewidth]{images/particles_left.pdf}
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\caption{Fraction of particles left in the Penning trap, with a given amplitude $f$.}
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\label{fig:particles_left}
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\end{figure}
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\end{document} |