From 0163bc3d21e26d238fa2879b4effd32c68f8f924 Mon Sep 17 00:00:00 2001 From: Janita Willumsen Date: Tue, 17 Oct 2023 19:41:01 +0200 Subject: [PATCH 1/2] Add introduction and (a somewhat messy) method part --- latex/appendix/appendix.tex | 44 +++++++++++++++++ latex/main.pdf | Bin 73209 -> 201809 bytes latex/main.tex | 31 +++++++----- latex/references/references.bib | 84 +++++++++++++++++++++++++++++--- latex/sections/abstract.tex | 4 +- latex/sections/conclusion.tex | 5 +- latex/sections/introduction.tex | 8 +-- latex/sections/methods.tex | 64 ++++++++++++++++++++++-- latex/sections/results.tex | 11 ++--- 9 files changed, 210 insertions(+), 41 deletions(-) create mode 100644 latex/appendix/appendix.tex diff --git a/latex/appendix/appendix.tex b/latex/appendix/appendix.tex new file mode 100644 index 0000000..eac7190 --- /dev/null +++ b/latex/appendix/appendix.tex @@ -0,0 +1,44 @@ +\documentclass[../main.tex]{subfiles} +\graphicspath{{\subfix{../images/}}} + +\begin{document} +\appendix +\section*{Appendix A}\label{sec:appendix_a} +Equations given +\begin{equation}\label{eq:newton_second} + m \ddot{\mathbf{r}} = \sum_{i} \mathbf{F}_{i} +\end{equation} +% + +% +\begin{equation}\label{eq:e_field_point} + \mathbf{E} = k_{e} \sum_{j=1}^{n} q_j \frac{\mathbf{r} - \mathbf{r}_{j}}{|\mathbf{r} - \mathbf{r}_{j}|^{3}} +\end{equation} +% +\begin{equation}\label{eq:e_field_potential} + \mathbf{E} = - \nabla V +\end{equation} + +\section*{Appendix B}\label{sec:appendix_b} +Sum of all forces +\begin{align*} + sum +\end{align*} + +We find the physical coordinates from $x(t) = \text{Re} f(t)$ and $y(t) = \text{Im} f(t)$, where $f(t)$ is given by equation \eqref{eq:general_solution}. + +We can rewrite $f(t)$ using the definition as +\begin{align*} + % f(t) =& A_{+}e^{-i(\omega_{+} t + \phi_{+})} + A_{-}e^{-i(\omega_{-} t + \phi_{-})} \\ + f(t) =& A_{+}(\cos{\omega_{+} t + \phi_{+}} - i \sin{\omega_{+} t + \phi_{+}}) \\ + \numberthis \label{eq:general_solution_trig} + &+ A_{-}(\cos{\omega_{-} t + \phi_{-}} - i \sin{\omega_{-} t + \phi_{-}}) +\end{align*} +% +If we rearrange the right side of equation \eqref{eq:general_solution_trig}, we find the physical coordinates +\begin{align}\label{eq:physical_coord} + x(t) &= A_{+}(\cos{\omega_{+} t + \phi_{+}}) + A_{-}(\cos{\omega_{-} t + \phi_{-}}) \\ + y(t) &= - A_{+}(i \sin{\omega_{+} t + \phi_{+}}) - A_{-}(i \sin{\omega_{-} t + \phi_{-}}) +\end{align} + +\end{document} \ No newline at end of file diff --git a/latex/main.pdf b/latex/main.pdf index ed5a52cc460302f0453e07df750681a66ab12161..c8b7b3aec771bcb9b0e2afd49fed7e1f304fa7e9 100644 GIT binary patch delta 167196 zcmY(KV{@Pl&}L)Xwrx*1(ZsfG8+UAHVrODwV%wV7w(Vq}w|2L-_Cr@)|KRF6efqTn zVy+AUl~VDi1S2aGCmiMM!q5sF7iS_WCMauKA}A;gaH+eQyv~&bJR1~rZa9yl%K8Rp zwYNKSIqfajWk*{$T+_+x&7u7m_#sUY{hRRk@ayqhA{e7Q@Ta-NK98D?HwjFT@E;-n zO-o@}3yntCr>Q?2x;>q~4+j|w?9*B$P3d~w8L(+=f6UcQ8Q7iO+p<4@f7Deo9Z67p 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zfd7BW^DDS&HXop#c0fUh5WN03fD=YPIie+}MXa>Z#aPI60R#d;VIgXSF0$mt`nHz% z>7kJH(ByOs&N|HFmaN)|g38KD*1Z>R>V@6|@j%@7(OkWl_yCR`8YbchP!$lX{8x8i5eD#$QOXaM@t<<(ZmW zf6$Rgr+7(iYHiKl)vrqEt@Pb~)e7?tcLN3>}&P diff --git a/latex/main.tex b/latex/main.tex index 7649808..4bd63f2 100644 --- a/latex/main.tex +++ b/latex/main.tex @@ -1,5 +1,5 @@ \documentclass[english,notitlepage,reprint,nofootinbib]{revtex4-1} % defines the basic parameters of the document -\usepackage{subfiles} +% % For preview: skriv i terminal: latexmk -pdf -pvc filnavn % If you want a single-column, remove "reprint" @@ -18,18 +18,18 @@ \usepackage{physics,amssymb} % mathematical symbols (physics imports amsmath) \usepackage{graphicx} % include graphics such as plots -\def\resources{\graphicspath{{"./images/"}}} +\graphicspath{{images/}} \usepackage{xcolor} % set colors \usepackage{hyperref} % automagic cross-referencing \usepackage{listings} % display code \usepackage{subfigure} % imports a lot of cool and useful figure commands % \usepackage{float} -%\usepackage[section]{placeins} +% \usepackage[section]{placeins} \usepackage{algorithm} \usepackage[noend]{algpseudocode} \usepackage{subfigure} \usepackage{tikz} -\usetikzlibrary{quantikz} +% \usetikzlibrary{quantikz2} % defines the color of hyperref objects % Blending two colors: blue!80!black = 80% blue and 20% black \hypersetup{ % this is just my personal choice, feel free to change things @@ -39,8 +39,12 @@ urlcolor={blue!80!black}} % Biblio stuff -\def\biblio{\bibliographystyle{plain}\bibliography{./references}} +% \def\biblio{\bibliographystyle{plain}\bibliography{../references/references}} +\newcommand\numberthis{\addtocounter{equation}{1}\tag{\theequation}} +\usepackage{xr} +\usepackage{subfiles} +% \externaldocument[M-]{\subfix{main}} \begin{document} @@ -50,28 +54,31 @@ \noaffiliation % ignore this, but keep it. % Abstract -\subfile{./sections/abstract.tex} +\subfile{sections/abstract} \maketitle % Introduction -\subfile{./sections/introduction.tex} +\subfile{sections/introduction} % Methods -\subfile{./sections/methods.tex} +\subfile{sections/methods} % Results -\subfile{./sections/results.tex} +\subfile{sections/results} % Conclusion -\subfile{./sections/conclusion.tex} +\subfile{sections/conclusion} + + +\subfile{appendix/appendix} \onecolumngrid -\bibliographystyle{plain} -\bibliography{.references/references} +\bibliographystyle{unsrt} +\bibliography{references/references} \end{document} diff --git a/latex/references/references.bib b/latex/references/references.bib index 236bbd0..b8d6413 100644 --- a/latex/references/references.bib +++ b/latex/references/references.bib @@ -1,9 +1,77 @@ - -@inbook{midpoint_rule, - author = "A.C Faul", - title = "A Concise Introduction to Numerical Analysis", - chapter = "5", - pages = "131", - publisher= "CRC Press", - year = "2016", +@book{vogel:2018, + author = {Manuel Vogel}, + title = {Particle Confinement in Penning Traps}, + subtitle = {An Introduction}, + publisher = {Springer Cham}, + year = {2018}, + edition = {1}, + abstract = {An introduction to the field of Penning traps and related experimental techniques. The book is motivated by the observation that often a vast number of different resources have to be explored to gain a good overview of Penning trap principles. This is especially true for students who experience additional difficulty due to the different styles of presentation and notation.} } + +@inbook{vogel:2018:ch5, + author = {Manuel Vogel}, + title = {Particle Confinement in Penning Traps}, + subtitle = {An Introduction}, + publisher = {Springer Cham}, + year = {2018}, + edition = {1}, + pages = {45--50} +} + +@inbook{vogel:2018:ch1, + author = {Manuel Vogel}, + title = {Particle Confinement in Penning Traps}, + subtitle = {An Introduction}, + publisher = {Springer Cham}, + year = {2018}, + edition = {1--6} +} + +@article{hunter:2007, + author = {Hunter, J. D.}, + title = {Matplotlib: A 2D graphics environment}, + journal = {Computing in Science \& Engineering}, + volume = {9}, + number = {3}, + pages = {90--95}, + abstract = {Matplotlib is a 2D graphics package used for Python for + application development, interactive scripting, and publication-quality + image generation across user interfaces and operating systems.}, + publisher = {IEEE COMPUTER SOC}, + doi = {10.1109/MCSE.2007.55}, + year = 2007 +} + +@inbook{lindstrom:2016:ch3, + author = {Tom Lindstrøm}, + title = {Kalkulus}, + publisher = {Universitetsforlaget}, + year = {2016}, + edition = {4}, + pages = {133--137} +} + +@inbook{lindstrom:2016:ch10:5, + author = {Tom Lindstrøm}, + title = {Kalkulus}, + publisher = {Universitetsforlaget}, + year = {2016}, + edition = {4}, + pages = {589--606} +} + +@article{britannica:2023:matter, + author = {The Editors of Encyclopaedia Britannica}, + title = {matter}, + year = {2023}, + url = {https://www.britannica.com/science/matter}, + urldate = {2023-10-17} +} + +@inbook{gjevik:2014:appendix, + author = {Bjørn Gjevik and Morten Wang Fagerland}, + title = {Feltteori og vektoranalyse}, + publisher = {Farleia Forlag}, + year = {2018}, + pages = {162--163} +} \ No newline at end of file diff --git a/latex/sections/abstract.tex b/latex/sections/abstract.tex index e170ec7..c7b5481 100644 --- a/latex/sections/abstract.tex +++ b/latex/sections/abstract.tex @@ -1,6 +1,5 @@ \documentclass[../main.tex]{subfiles} -\resources % search for graphics in ../res/imgs/ - +\graphicspath{{\subfix{../images/}}} \begin{document} @@ -8,5 +7,4 @@ Add an abstract for the project? \end{abstract} -\biblio \end{document} \ No newline at end of file diff --git a/latex/sections/conclusion.tex b/latex/sections/conclusion.tex index 4107c03..10a537a 100644 --- a/latex/sections/conclusion.tex +++ b/latex/sections/conclusion.tex @@ -1,6 +1,5 @@ \documentclass[../main.tex]{subfiles} -\resources % search for graphics in ../res/imgs/ - +\graphicspath{{\subfix{../images/}}} \begin{document} @@ -9,5 +8,5 @@ -\biblio + \end{document} \ No newline at end of file diff --git a/latex/sections/introduction.tex b/latex/sections/introduction.tex index da0b77f..0c4b1be 100644 --- a/latex/sections/introduction.tex +++ b/latex/sections/introduction.tex @@ -1,13 +1,13 @@ \documentclass[../main.tex]{subfiles} -\resources % search for graphics in ../res/imgs/ - +\graphicspath{{\subfix{../images/}}} \begin{document} \section{Introduction} +We are surrounded by matter, which are made up of elementary particles. In the field of physics we want to understand the properties of these particles, measure their physical quantities, not to mention explain the origin of mass \cite{britannica:2023:matter}. +However, to study a particle, it is necessary isolate and contain it over time. The Penning trap is a device, able to confine charged particles for a period of time. This concept was evolved from F. M. Penning's implementation of magnetic fiels to a vaccum gauge, and J. R. Pierce's work with electron beams, and put into practice by Hans Dehmelt. In 1973 Dehmelt and his group of researchers were able to contain a particle and store it over several months \cite{vogel:2018:ch1}. +We want to study the effects of the Penning trap, when it is used to confine charged particles. Our focus will be on an ideal Penning trap, where an electrostatic field confines the particle in z-direction, and a magnetic field confines it in the radial direction. Using a numerical approach we will model a single particle, to study the particle motion in radial direction. In addition, we will model a system of particles, to study their motion both with and without particle interaction. - -\biblio \end{document} \ No newline at end of file diff --git a/latex/sections/methods.tex b/latex/sections/methods.tex index e2f6d07..409b5bc 100644 --- a/latex/sections/methods.tex +++ b/latex/sections/methods.tex @@ -1,13 +1,69 @@ \documentclass[../main.tex]{subfiles} -\resources % search for graphics in ../res/imgs/ - +\graphicspath{{\subfix{../images/}}} \begin{document} - \section{Methods} +% problem 1 +When we study the Penning traps effect on a particle with a charge $q$, we need to consider the forces acting on the particle. The sum of all forces acting on the particle, is given by the Lorentz force \eqref{eq:lorentz_force}. +\begin{equation}\label{eq:lorentz_force} + \mathbf{F} = q \mathbf{E} + q \mathbf{v} \times \mathbf{B}, +\end{equation} +We can use Newton's second law \eqref{eq:newton_second} to determine this sum by +\begin{align*} + \ddot{\mathbf{r}} &= \frac{1}{m} \sum_{i} \mathbf{F}_{i} \\ + &= \frac{1}{m} (q \mathbf{E} + q \mathbf{v} \times \mathbf{B}) \\ + &= \frac{q}{m} \big(\frac{V_{0}}{d^{2}} (x, y, -2z) + (B_{0}\dot{y}, -B_{0}\dot{x}, 0) \big), +\end{align*} +where the complete derivation can be found in \ref{sec:appendix_b}. We can now write the particle's position as +\begin{align} + \label{eq:motion_x} + \ddot{x} - \omega_{0} \dot{y} - \frac{1}{2} \omega_{z}^{2} x &= 0, \\ + \label{eq:motion_y} + \ddot{y} + \omega_{0} \dot{x} - \frac{1}{2} \omega_{z}^{2} y &= 0, \\ + \label{eq:motion_z} + \ddot{z} + \omega_{z}^{2} z &= 0, +\end{align} +% define w_0 og w_z i appendix? +In addition, we can find the general solution for eq. \eqref{eq:motion_z}, when we consider the characteristic equation of a second order differential equation \cite{lindstrom:2016:ch10:5}. +\begin{align*} + r^{2} + \omega_{z}^{2} &= 0 \\ + r &= \pm \sqrt{- \omega_{z}^{2}} = \mp i \omega_{z}, +\end{align*} +two complex roots which gives us solutions in the form of +\begin{equation}\label{eq:all_diff_sol} + z = c_{1} e^{i \omega_{z} t} + c_{2} e^{-i \omega_{z} t}, +\end{equation} +For a complex number $z = a + ib$, we can define $e^{z} \equiv e^{a} (\cos{b} + i \sin{b})$ \cite{lindstrom:2016:ch3}. We can rewrite \eqref{eq:all_diff_sol} as +\begin{align*} + c_{1} e^{i \omega_{z} t} + c_{2} e^{-i \omega_{z} t} &= c_{1} (\cos{\omega_{z} t} + i \sin{\omega_{z} t}) + c_{2} (\cos{\omega_{z} t} - i \sin{\omega_{z} t} \\ + &= E \cos{\omega_{z} t} + i F \sin{\omega_{z} t} +\end{align*} +% +Since \eqref{eq:motion_x} and \eqref{eq:motion_y} are coupled, we want to rewrite it as a single differential equation. We can obtain this by introducing $f(t) = x(t) + iy(t)$, +\begin{align*} + (\ddot{x} - \omega_{0} \dot{y} - \frac{1}{2} \omega_{z}^{2} x) + i (\ddot{y} + \omega_{0} \dot{x} - \frac{1}{2} \omega_{z}^{2} y) &= 0 \\ + \ddot{x} - \omega_{0} \dot{y} - \frac{1}{2} \omega_{z}^{2} x + i\ddot{y} + i\omega_{0} \dot{x} - i \frac{1}{2} \omega_{z}^{2} y &= 0 \\ + \ddot{x} + i\ddot{y} + i\omega_{0} \dot{x} - \omega_{0} \dot{y} - \frac{1}{2} \omega_{z}^{2} x - i \frac{1}{2} \omega_{z}^{2} y &= 0 \\ + \ddot{x} + i\ddot{y} + i\omega_{0} (\dot{x} + i \dot{y}) - \frac{1}{2} \omega_{z}^{2} x - i \frac{1}{2} \omega_{z}^{2} y &= 0 \text{where $i \omega_{0} \dot{x} + (-1) \omega_{0} \dot{y} = i \omega_{0} \dot{x} + i^{2} \omega_{0} \dot{y}$} \\ + \ddot{f} + i \omega_{0} \dot{f} - \frac{1}{2} \omega_{z}^{2} f &= 0 +\end{align*} + + -\biblio + +Physical properties given by newtons second law \eqref{eq:newton_second} +\begin{equation}\label{eq:general_solution} + f(t) = A_{+}e^{-i(\omega_{+} t + \phi_{+})} + A_{-}e^{-i(\omega_{-} t + \phi_{-})} +\end{equation} +The particle moves and its position can be determined using newton. where the electric field + +\subsection*{Algorithm} + + +\subsection*{Tools} +We used matplotlib + \end{document} \ No newline at end of file diff --git a/latex/sections/results.tex b/latex/sections/results.tex index 003cba3..c48a6b3 100644 --- a/latex/sections/results.tex +++ b/latex/sections/results.tex @@ -1,13 +1,10 @@ \documentclass[../main.tex]{subfiles} -\resources % search for graphics in ../res/imgs/ - +\graphicspath{{\subfix{../images/}}} \begin{document} -\section{Results} +\section{Results and Discussion} +From equation \eqref{eq:general_solution} we can find - - - -\biblio +Problem 9: we have to deal with $|\mathbf{r}| = d$ \end{document} \ No newline at end of file From d07e5a9effe412fdaa36c1256331216adadf4417 Mon Sep 17 00:00:00 2001 From: Janita Willumsen Date: Wed, 18 Oct 2023 13:31:45 +0200 Subject: [PATCH 2/2] Update introduction --- 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