156 lines
6.2 KiB
TeX
156 lines
6.2 KiB
TeX
% USEFUL LINKS:
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% -------------
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% - UiO LaTeX guides: https://www.mn.uio.no/ifi/tjenester/it/hjelp/latex/
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% - Mathematics: https://en.wikibooks.org/wiki/LaTeX/Mathematics
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% - Physics: https://ctan.uib.no/macros/latex/contrib/physics/physics.pdf
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% - Basics of Tikz: https://en.wikibooks.org/wiki/LaTeX/PGF/Tikz
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% - All the colors! https://en.wikibooks.org/wiki/LaTeX/Colors
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% - How to make tables: https://en.wikibooks.org/wiki/LaTeX/Tables
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% - Code listing styles: https://en.wikibooks.org/wiki/LaTeX/Source_Code_Listings
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% - \includegraphics https://en.wikibooks.org/wiki/LaTeX/Importing_Graphics
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% - Learn more about figures: https://en.wikibooks.org/wiki/LaTeX/Floats,_Figures_and_Captions
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% - Automagic bibliography: https://en.wikibooks.org/wiki/LaTeX/Bibliography_Management (this one is kinda difficult the first time)
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% (This document is of class "revtex4-1", the REVTeX Guide explains how the class works)
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% REVTeX Guide: http://www.physics.csbsju.edu/370/papers/Journal_Style_Manuals/auguide4-1.pdf
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% COMPILING THE .pdf FILE IN THE LINUX TERMINAL
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% ---------------------------------------------
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%
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% [terminal]$ pdflatex report_example.tex
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% Run the command twice, always.
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% When using references, footnotes, etc. you should run the following chain of commands:
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% [terminal]$ pdflatex report_example.tex
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% [terminal]$ bibtex report_example
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% [terminal]$ pdflatex report_example.tex
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% [terminal]$ pdflatex report_example.tex
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%
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% This series of commands can of course be gathered into a single-line command:
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% [terminal]$ pdflatex report_example.tex && bibtex report_example.aux && pdflatex report_example.tex && pdflatex report_example.tex
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%
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% ----------------------------------------------------
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% \documentclass[english,notitlepage,reprint,nofootinbib]{revtex4-2} % defines the basic parameters of the document
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\documentclass[english,notitlepage,reprint,nofootinbib]{revtex4-2} % defines the basic parameters of the document
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% If you want a single-column, remove "reprint"
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\usepackage{silence}
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\WarningFilter{revtex4-2}{Repair the float}
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% Allows special characters (including æøå)
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\usepackage[utf8]{inputenc}
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\usepackage[english]{babel}
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% Note that you may need to download some of these packages manually, it depends on your setup.
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% It may be usefult to download TeXMaker, because it includes a large library of the most common packages.
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\usepackage{amssymb} % mathematical symbols (physics imports amsmath)
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\usepackage{amsmath}
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\usepackage{graphicx} % include graphics such as plots
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\usepackage{xcolor} % set colors
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\usepackage{hyperref} % automagic cross-referencing
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\usepackage{listings} % display code
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\usepackage{subfigure} % imports a lot of cool and useful figure commands
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% \usepackage{float}
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%\usepackage[section]{placeins}
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\usepackage{algorithm}
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\usepackage[noend]{algpseudocode}
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\usepackage{subfigure}
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\usepackage{tikz}
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\usepackage{cleveref}
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\usepackage{siunitx}
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\usetikzlibrary{quantikz}
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% defines the color of hyperref objects
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% Blending two colors: blue!80!black = 80% blue and 20% black
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\hypersetup{ % this is just my personal choice, feel free to change things
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colorlinks,
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linkcolor={red!50!black},
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citecolor={blue!50!black},
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urlcolor={blue!80!black}}
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% ===========================================
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\newcommand{\vstat}{V_\mathrm{stat.}}
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\newcommand{\vdyn}{V_\mathrm{dyn.}}
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\newcommand{\estat}{E_\mathrm{stat.}}
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\newcommand{\edyn}{E_\mathrm{dyn.}}
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\newcommand{\estatc}[1]{E_{\mathrm{stat.},#1}}
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\newcommand{\edync}[1]{E_{\mathrm{dyn.},#1}}
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\begin{document}
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\title{Accuracy and Stability of Numerical Integration Methods in Penning Trap Simulations} % self-explanatory
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\author{Lars Bogner} % self-explanatory
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\date{\today} % self-explanatory
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\noaffiliation % ignore this, but keep it.
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%This is how we create an abstract section.
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\begin{abstract}
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\textcolor{red}{TODO: ABSTRACT HERE.}
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\end{abstract}
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\maketitle
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% ===========================================
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\section{Introduction}
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\input{chapters/introduction}
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% ===========================================
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\section{Methods}\label{sec:methods}
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%
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\input{chapters/methods}
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% ===========================================
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\section{Results and discussion}\label{sec:results_and_discussion}
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%
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\input{chapters/results}
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% ===========================================
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\section{Conclusion}\label{sec:conclusion}
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%
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\input{chapters/conclusion}
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% ===========================================
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\appendix
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\section{Electric Field Equations} \label{app:efield_equations}
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Here we derive the electric field equations used in \cref{sec:methods}. The electric field following from $\vstat$ defined in \cref{eq:V_stat} is given by
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\begin{align}
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\estatc{x} &= -\frac{\partial \vstat}{\partial x} = \frac{V_0}{d^2} x, \\
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\estatc{y} &= -\frac{\partial \vstat}{\partial y} = \frac{V_0}{d^2} y, \\
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\estatc{z} &= -\frac{\partial \vstat}{\partial z} = -\frac{2 V_0}{d^2} z,
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\end{align}
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for all $|\vec r| < d$ and zero otherwise. The electric field following from $\vdyn$ is simply
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\begin{equation}
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\edyn = (1 + f \cos \omega_V t) \cdot \estat.
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\end{equation}
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\section{Special Case Analytical Solution} \label{app:special_case_analytical_solution}
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In the special case of a single particle in the potential $\vstat$, with the inital conditions $\vec{r}(t_0) = (x_0, 0, z_0)$ and $\vec{v}(t_0) = (0, v_0, 0)$, we can derive an analytical solution for the trajectory. The solution for the $z$ component is straightforward
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\begin{equation}
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z(t) = z_0 \cos(\omega_z t).
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\end{equation}
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From the equation system
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\begin{align}
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f(0) = A_+ e^{-i \phi_+} + A_- e^{-i \phi_-} &\equiv x_0, \\
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\dot f(0) = -i \omega_+ A_+ e^{-i \phi_+} - i \omega_- A_- e^{-i \phi_-} &\equiv i v_0,
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\end{align}
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we derive $\phi_\pm = 0$ as $f(0) \in \mathbb{R}$ and therefore $A_+ + A_- = x_0$ and $\omega_+ A_+ + \omega_- A_- = -v_0$. Solving this system of equations for $A_\pm$ we find
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\begin{equation}
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A_\pm =\pm \frac{v_0 + \omega_\mp x_0}{\omega_- - \omega_+}.
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\end{equation}
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% ===========================================
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\onecolumngrid
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% \bibliographystyle{apalike}
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\bibliographystyle{unsrt}
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\bibliography{ref}
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\end{document} |