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zhBl9+5h{!bY5W(3vM9Hk>JAQ`&yJcLKdGw6#KgoKu0^)*E_xo~%59{hbGZ-ijz!~j zwdkcL`+4(Pzw!8*JY&)8LS{)1QD#c|zv1&ydw=r6n~GowxsMydm1tAdsB&v_n~n#H zn$Nx|uoWsl0Th-h=H8kM)9;wb@pefM<}o@L)Ll&mgwxUB;1fD@4gWNcH{$Ou!FW#e`l4oc%yr`YdCSbABNFar-&nmv>j!RMZmfpDG zo8_D}yd{)M7xVan#_I=jtSt@dXZa z;1-n0BhMF;>N#1x0GyhVK3w{^JddO`Bx$#Nu*4nqJJ^4KUhto5Tt_gakNFLV~~EodDkDL z@NbUw|G}IRxYA#|>3^R!{l$|)ep29n5~cg}B&3ADf2QyQOAJZsKSBP%um8mkLw@tu zf8l zru#@_f|KnW;R3<|Ud{zpc<6o?Pd z?JpUWaC`hs27wTUFn^N~dh|~jjBpzNh7$)9PW|6y;{VhYmmrLm{wod&gZ;xVR0{kr zGBLuh{J-i-fMJCD;%_nt_#ZwbAfo?*6aObpoOnt8&7Qc}A9;9Tkd7`W%#Tr{fulcR zjzpLjX}Z~?fj Date: Tue, 24 Oct 2023 17:21:27 +0200 Subject: [PATCH 02/23] Update sections --- latex/sections/abstract.tex | 4 +- latex/sections/conclusion.tex | 7 +- latex/sections/introduction.tex | 8 +- latex/sections/methods.pdf | Bin 254541 -> 0 bytes latex/sections/methods.tex | 279 ++++++++++++++------------------ latex/sections/results.tex | 148 +++++++++-------- 6 files changed, 215 insertions(+), 231 deletions(-) delete mode 100644 latex/sections/methods.pdf diff --git a/latex/sections/abstract.tex b/latex/sections/abstract.tex index c7b5481..cb358a8 100644 --- a/latex/sections/abstract.tex +++ b/latex/sections/abstract.tex @@ -4,7 +4,9 @@ \begin{document} \begin{abstract} - Add an abstract for the project? + We have studied the motion of singly-charged Calcium ions ($\text{Ca}^{+}$), inside an ideal Penning trap. With a numerical approach, studied the equations of motion by implementing the forward Euler method \(FE\) and the 4th order Runge-Kutta \(RK4\). We found that RK4 approximates the solution with smaller relative error than FE. In addition, we evaluated methods by rate of convergence. We found that RK4 has a higher convergence rate at approx. $4.0$, compared to FE at approx. $1.4$. + + Freq. \end{abstract} \end{document} \ No newline at end of file diff --git a/latex/sections/conclusion.tex b/latex/sections/conclusion.tex index 10a537a..9a72d93 100644 --- a/latex/sections/conclusion.tex +++ b/latex/sections/conclusion.tex @@ -2,11 +2,10 @@ \graphicspath{{\subfix{../images/}}} \begin{document} - \section{Conclusion} +We studied the movement of particles confined by an ideal Penning trap, where we used iterative methods to simulate the particle behavior. We included the magnetic and electric field of the Penning trap, in addition to simulating the particles behavior when interaction with other each other. When we introduced the interaction, the movement in both radial direction and z-direction changed. From a circular path, to a more elliptical path, where the particles initial condition determine how it is affecting other particles path. +We also compared iterative methods with the analytical solution, and found that the forward Euler \(FE\) method result in an approximation with larger relative error than the 4th order Runge-kutta \(RK4\) method. In addition to a small relative error, we also found that RK4 has a higher convergence rate at approx. $4.0$, compared to FE at approx. $1.4$. Which suggest RK4 ... - - - +Resonance and freq? \end{document} \ No newline at end of file diff --git a/latex/sections/introduction.tex b/latex/sections/introduction.tex index 0ef8ecb..5a3cbfc 100644 --- a/latex/sections/introduction.tex +++ b/latex/sections/introduction.tex @@ -4,10 +4,12 @@ \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}. +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, and measure their physical quantities. We want to describe the particles such that we can 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 confine a particle and store it over several months \cite{vogel:2018:ch1}. +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 fields to a vaccuum 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 confine a particle and store it over several months \cite{vogel:2018:ch1}. -In practice, a Penning trap is not easy to obtain, and an experiment is both time consuming and expensive. A numerical approach, allow us to study the effects of the Penning trap on a charged particle, without the cost. We can use ordinary differential equations to model the particle's movement, confined within an Penning trap. 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. We will use numerical methods to model a single particle, and study the particle motion in radial direction. In addition, we will model a system of particles, and study their motion both with and without particle interaction. +In practice, a Penning trap is not easy to obtain, and an experiment would be both time consuming and expensive. A numerical approach, allow us to study the effects of the Penning trap on a charged particle, without the equipment and material cost. With ordinary differential equations (ODE) we can model the movement of particles, confined within a Penning trap. + +We will study an ideal Penning trap, where an electrostatic field confines the particle in z-direction, and a magnetic field confines it in the radial direction. We will use numerical methods to model a single particle, and study the particle motion in radial direction. In addition, we will model a system of particles, and study their motion both with and without particle interaction. % \end{document} \ No newline at end of file diff --git a/latex/sections/methods.pdf b/latex/sections/methods.pdf deleted file mode 100644 index abf8b57fd9798f8165afd65ea3ec2f82953d3b4a..0000000000000000000000000000000000000000 GIT binary patch literal 0 HcmV?d00001 literal 254541 zcma%?Lv${HvSwr3wr$(CZR^C&AM?bvZQHi(oY;2Kw|n&)-N~y_)v#u}cCC+GNlb#C znSmXKe13Uk9fpgDiOA8!28NFhhEdMk!P3==h@F$0mFWK!45Ng#ovXPs5u=2iv8%b5 zxv8U>IgEe+jEk$Yxv@Qr=jICkMj~0;pW9FLYszPc4f5^q)#RYD#X`|kGiNz&5f!e? z9d=&vSLQ9q@GkGWI$l&Eb>NlqB1?<3mnj8(crp|(OOmwd-RbcwLj&JGkB4i9tjtZJ z>T)v<4wfnxK`H`EDi{(2MbB4tE*}rKoWCDIA%rUd0pIMHte|~rAORgnYUq;>8$lC; zC^rEg^7qcd6$g(2K7Nm%f_~5VvCA3G1?t4b?tQA+AA8@&wc9;L5J6<%){L-EJcr`8`kaBlJ5glK{)xF^IYZOv+x%GxXc65?y+AYpaw&TE0 z9X0?pHsC(2d&%|UI`!{{1~|-!hSjOO0e=KKDeZp79uAuKzAw%Q31tDc*L!xU?VyNG zq|0B9VkmtM14VAEm9V2$Jr8@|48gcEFhduNMtB!xAY$$4%oZ>tR-MIeUNneXHjqLr 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The electric potential $V_{0}$ is applied to the electrodes, where the end caps are positively charged and the ring is negatively charged. The particle is confined by the electric field in the z-direction, however, it is not confined in the radial direction (xy-plane). The magnetic field is necessary to ensure the particle is fully confined in the Penning trap, and will force the particle to move in a circular orbit. + +First, we consider Newton's second law \eqref{eq:newton_second}, to determine the position of the particle. +\begin{equation}\label{eq:newton_second} + m \ddot{\mathbf{r}} = \sum_{i} \mathbf{F}_{i} +\end{equation} % +% +In addition, we introduce the Lorentz force \eqref{eq:lorentz_force}, which describes the force acting the particle. +\begin{equation}\label{eq:lorentz_force} + \mathbf{F} = q \mathbf{E} + q \mathbf{v} \times \mathbf{B}, +\end{equation} % +% +where $\ddot{\mathbf{r}} = d^{2} \mathbf{r} / dt^{2}$. We combine eq. \eqref{eq:newton_second} and eq. \eqref{eq:lorentz_force} in \begin{equation}\label{eq:newton_lorentz} - m \ddot{\mathbf{r}} = (q \mathbf{E} + q \mathbf{v} \times \mathbf{B}). -\end{equation} + m \ddot{\mathbf{r}} = (q \mathbf{E} + q \mathbf{v} \times \mathbf{B}), +\end{equation} % % -Using eq. \eqref{eq:newton_lorentz} we derive the differential equations in sec. \ref{sec:eq_motion}, and can rewrite them as +and derive the differential equations in appendix \ref{sec:eq_motion}. We can rewrite and simplify these equations as \begin{align} \label{eq:motion_x} \ddot{x} - \omega_{0} \dot{y} - \frac{1}{2} \omega_{z}^{2} x &= 0, \\ @@ -20,32 +53,36 @@ Using eq. \eqref{eq:newton_lorentz} we derive the differential equations in sec. \ddot{z} + \omega_{z}^{2} z &= 0, \end{align} % % +where +\begin{equation*} + \omega_{0} \equiv \frac{q B_{0}}{m}, \quad \omega_{z}^{2} \equiv \frac{2 q V_{0}}{m d^{2}}. +\end{equation*} % +% We find the general solution for eq. \eqref{eq:motion_z} \begin{equation}\label{eq:eq_general} z(t) = c_{1} e^{i \omega_{z} t} + c_{2} e^{-i \omega_{z} t}, \end{equation} % -derived in sec. \ref{sec:eq_general}. Continuing, we will use a Calcium ion with a single positive charge. That is, we assume the charge of the particle is $q > 0$. +derived in appendix \ref{sec:eq_general}. Continuing, we will use a Calcium ion with a single positive charge. That is, we assume the charge of the particle is $q > 0$. -% Problem 2 -Since eq. \eqref{eq:motion_x} and eq. \eqref{eq:motion_y} are coupled, we want to rewrite them as a single differential equation. We derive this in sec. \ref{sec:eq_complex}, and the resulting equation is given by +Since eq. \eqref{eq:motion_x} and eq. \eqref{eq:motion_y} are coupled, we want to rewrite them as a single differential equation. We derive this in appendix \ref{sec:eq_complex}, and the resulting equation is given by \begin{equation}\label{eq:single_differential} \ddot{f} + i \omega_{0} \dot{f} - \frac{1}{2} \omega_{z}^{2} f = 0. -\end{equation} - -% Problem 3 +\end{equation} % +% Eq. \eqref{eq:single_differential} has a general solution given by \begin{equation}\label{eq:general_solution} f(t) = A_{+}e^{-i(\omega_{+} t + \phi_{+})} + A_{-}e^{-i(\omega_{-} t + \phi_{-})}. \end{equation} % -The amplitude $A_{+}$ and $A_{-}$ are positive, the phases $\phi_{+}$ and $\phi_{-}$ are constant, and the rate is given by -\begin{equation*} +% +The amplitude $A_{+}$ and $A_{-}$ are positive, the phases $\phi_{+}$ and $\phi_{-}$ are constant, and the angular rate is given by +\begin{equation}\label{eq:angular_rate} \omega_{\pm} = \frac{\omega_{0} \pm \sqrt{\omega_{0}^{2} - 2 \omega_{z}^{2}}}{2}. -\end{equation*} +\end{equation} We find the physical coordinates at a given time $t$ using \begin{equation*} x(t) = \text{Re}f(t), \quad y(t) = \text{Im}f(t), \end{equation*} % -and eq. \eqref{eq:general_solution}. We can rearrange the right hand side of the derived expression in \ref{sec:eq_coord}, and find the physical coordinates +and eq. \eqref{eq:general_solution}. We can rearrange the right hand side of the expression derived in \ref{sec:eq_coord}, to 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_{+} \sin(\omega_{+} t + \phi_{+}) - A_{-} \sin(\omega_{-} t + \phi_{-}) @@ -55,104 +92,74 @@ However, to obtain a bound on the particle's movement in the radial direction, w \begin{align*} |f(t)| &= \sqrt{(x(t))^{2} + (y(t))^{2}}. \\ \end{align*} % +Which means we have to put constraint on the values of $\omega_{0}$ and $\omega_{z}^{2}$ in order to find a bounded solution $|f(t)| < \infty$. We derive an expression for this in appendix \ref{sec:bounded_solution}. Since we have assumed the particle charge $q > 0$, and the mass $m > 0$, the constraints are +\begin{equation} + \frac{q}{m} > \frac{4 V_{0}}{d^{2} B_{0}^{2}}. +\end{equation} % +% When $t \rightarrow \infty$, the upper and lower limits are \begin{align} - R_{+} &= A_{+} + A_{-} \\ + \label{eq:upper_b} + R_{+} &= A_{+} + A_{-}, \\ + \label{eq:lower_b} R_{-} &= |A_{+} - A_{-}|, \end{align} -derived in sec. \ref{sec:bounded_solution}. +which we derive in appendix \ref{sec:upper_lower_bound}. -% Problem 3 -To obtain the bounded solution, we need to consider the expression +In this experiment, we will use the initial conditions for the particle given in table \ref{tab:initial_particle_cond}. From eq. \eqref{eq:motion_z} and the initial conditions, we find the specific solution \begin{equation*} - \alpha = (\omega_{+} - \omega_{-}) t +( \phi_{+} - \phi_{-}). + z(t) = z_{0} \cos (\omega_{z} t) \end{equation*} -When $t \rightarrow \infty$ the constant phases will not affect the expression, we obtain a bounded solution if we consider the rate such that $|f(t)| < \infty$. That is, we need $\omega_{0}^{2} - 2 \omega_{z}^{2}$ in eq. \eqref{eq:angular_rate} to be real. +For the particle movement in the radial direction, we find the specific solution using eq. \eqref{eq:general_solution} with \begin{align*} - \omega_{0}^{2} - 2 \omega_{z}^{2} = 0, \quad \omega_{0} > 2 \omega_{z} + A_{+} &= \frac{v_{0} + \omega_{-} x{0}}{\omega_{-} - \omega_{+}}, & A_{-} &= - \frac{v_{0} + \omega_{+} x{0}}{\omega_{-} - \omega_{+}}, \\ + \phi_{+} &= 0, & \phi_{-} &= 0. \end{align*} -\begin{equation}\label{eq:angular_rate} - \omega_{\pm} = \frac{\omega_{0} \pm \sqrt{\omega_{0}^{2} - 2 \omega_{z}^{2}}}{2} + +We will also consider the numerical simulation of multiple particles, confined in a Penning trap. The particles will experience a repelling force from each other, giving the electric field +\begin{equation}\label{eq:electric_field_interaction} + \mathbf{E} = k_{e} \sum_{j = 1}^{n} q_j \frac{\mathbf{r} - \mathbf{r}_{j}}{|\mathbf{r}_{i} - \mathbf{r}_{j}|^{3}}. +\end{equation} % +The Coulomb constant $k_{e}$ value can be found in table \ref{tab:constants}. + +For multiple particles we have to modify the equations of motion, by adding a term for the force a given particle experience at a given point. When we scale eq. \eqref{eq:electric_field_interaction} by charge an mass, we get a new set of equations of motion +\begin{align} + \label{eq:coulomb_motion_x} + \ddot{x}_{i} - \omega_{0,i} \dot{y}_{i} - \frac{1}{2} \omega_{z,i}^{2} x_{i} - k_{e} \frac{q_{i}}{m_{i}}\sum_{j \neq i} q_j \frac{x_{i} - x_{j}}{|\mathbf{r}_{i} - \mathbf{r}_{j}|^{3}} &= 0, \\ + \label{eq:coulomb_motion_y} + \ddot{y}_{i} - \omega_{0,i} \dot{x}_{i} - \frac{1}{2} \omega_{z,i}^{2} y_{i} - k_{e} \frac{q_{i}}{m_{i}}\sum_{j \neq i} q_j \frac{y_{i} - y_{j}}{|\mathbf{r}_{i} - \mathbf{r}_{j}|^{3}} &= 0, \\ + \label{eq:coulomb_motion_z} + \ddot{z}_{i} + \omega_{z,i}^{2} z_{i} - k_{e} \frac{q_{i}}{m_{i}}\sum_{j \neq i} q_j \frac{z_{i} - z_{j}}{|\mathbf{r}_{i} - \mathbf{r}_{j}|^{3}} &= 0, +\end{align} % +where $i$ and $j$ denote the particle indices. When we include a time-dependence to the applied potential we make a replacement of the initial electric potential +\begin{equation}\label{eq:pertubation} + V_{0} \rightarrow V_{0} (1 + f \cos (\omega_{V} t)), \end{equation} +where $f$ denotes the amplitude and $\omega_{V}$ the angular rate. -We find the solution -Physical properties given by newtons second law \eqref{eq:newton_second} - -The particle moves and its position can be determined using newton. where the electric field - - - - - -\begin{table}[H] - \centering - \begin{tabular}[c]{lll} - Constant & Value & \\ - \hline - $B_{0}$ & $1.00 T$ & $9.65 \cross 10^{1} \frac{u}{(\mu s) e}$ \\ - $V_{0}$ & $25.0 mV$ & $2.41 \cross 10^{6} \frac{u (\mu m)^{2}}{(\mu s)^{2} e}$ \\ - $d$ & $500 \ \mu m$ & \\ - \hline - \end{tabular} - \caption{Default configuration of the Penning trap, where the value of T and V can be found in table \ref{tab:constants}.} - \label{tab:penning_config} -\end{table} - -\subsection{Dealing with a multi--particle system} - -For a multi-particles system, we need to modify $\vb{F}$ to account for the -force of other particles in the system acting upon each other. To do that, we -add another term to $\vb{F}$ -\begin{equation} - \vb{F}_i(t, \vb{v}_i, \vb{r}_i) = q_i \vb{E}(t, \vb{r}_i) + q_i \vb{v}_i \cross \vb{B} - \vb{E}_p(t, \vb{r}_i), -\end{equation} -where $i$ and $j$ are particle indices and -\begin{equation} - \vb{E}_p(t, \vb{r}_i) = q_i k_e \sum_{j \neq i} - q_j \frac{\vb{r_i} - \vb{r_j}}{\left| \vb{r_i} - \vb{r_j} \right|^3}. - \label{eq:} +\subsection{Algorithms and implementation}\label{sec:algo_implementation} +When we consider a multi-particle system, we adapt the notation in our derived equations. The force acting on a particle in the Penning trap, which experience particle interaction, can be written as +\begin{equation}\label{eq:lorentz_force_interaction} + \mathbf{F}_{i}(t, \mathbf{v}_{i}, \mathbf{r}_{i}) = q_{i} \mathbf{E}(t, \mathbf{r}_{i}) + q_{i} \mathbf{v}_{i} \cross \mathbf{B} - \mathbf{E}_{p}(t, \mathbf{r}_{i}), \end{equation} +where $i$ and $j$ still denotes the particle indices, and $\mathbf{E}_{p}$ denotes the force exerted on particle $i$ by particle $j$. +\begin{equation}\label{eq:electric_force_interaction} + \mathbf{E}_{p}(t, \mathbf{r}_{i}) = k_{e} q_{i} \sum_{j \neq i} + q_{j} \frac{\mathbf{r}_{i} - \mathbf{r}_{j}}{| \mathbf{r}_{i} - \mathbf{r}_{j} |^{3}}. +\end{equation} % +In addition, we adapt the notation of \eqref{eq:newton_second}, and define the first derivative of the particle position as +\begin{align} + \label{eq:r_derivative} + \frac{d \mathbf{r}_{i}}{dt} = \dot{\mathbf{r}}_{i} &= \mathbf{v}_{i}, \\ + \label{eq:v_derivative} + \frac{d \mathbf{v}_{i}}{dt} = \dot{\mathbf{v}}_{i} &= \frac{\mathbf{F}_{i} (t, \mathbf{v}_{i}, \mathbf{r}_{i}}{m_{i}}, +\end{align} -Newton's second law for a particle $i$ is then -\begin{equation} - \frac{d^2\vb{r}_i}{dt^2} = \frac{\vb{F}_i\left(t, \frac{d\vb{r}_i}{dt}, \vb{r_i}\right)}{m_i}, - \label{eq:newtonlaw2} -\end{equation} -We can then rewrite the second order ODE from equation~\ref{eq:newtonlaw2} -into a set of coupled first order ODEs. -We now rewrite Newton's second law of motion as -\begin{equation} - \begin{split} - \frac{d\vb{r}_i}{dt} &= \vb{v}_i \\ - \frac{d\vb{v}_i}{dt} &= \frac{\vb{F}_i(t, \vb{v}_i, \vb{r}_i)}{m_i}. - \end{split} - \label{eq:coupled} -\end{equation} - -\subsection{Forward Euler} - -For a particle $i$, the forward Euler method for a coupled system is -expressed as -\begin{equation} - \begin{split} - \vb{r}_{i,j+1} &= \vb{r}_{i,j} + h \cdot \frac{d\vb{r}_{i,j}}{dt} = \vb{r}_{i,j} + h \cdot \vb{v}_{i,j} \\ - \vb{v}_{i,j+1} &= \vb{v}_{i,j} + h \cdot \frac{\vb{v}_{i,j}}{dt} = \vb{v}_{i,j} + h \cdot \frac{\vb{F}\left( t_{j}, \vb{v}_{i,j}, \vb{r}_{i,j} \right)}{m}, - \end{split} -\end{equation} -for particle $i$ where $j$ is the current time step of the particle, -$m$ is the mass of the particle, and $h$ is the step length. - -When dealing with a multi-particle system, we need to ensure that we do not -update the position of any particles until every particle has calculated their -next step. An easy way of doing this is to create a copy of all the particles, -then update the copy, and when all the particles have calculated their next -step, simply replace the particles with the copies. -Algorithm~\ref{algo:forwardeuler} provides an overview on how that can be achieved. % Make this better - +We first implemented the forward Euler method, using the expression for a coupled system given in appendix \ref{sec:algo_euler}. We define the forward Euler algorithm in \ref{algo:forward_euler}. \begin{figure}[H] \begin{algorithm}[H] \caption{Forward Euler method} - \label{algo:forwardeuler} + \label{algo:forward_euler} \begin{algorithmic} \Procedure{Evolve forward Euler}{$particles, dt$} \State $N \leftarrow \text{Number of particles in } particles$ @@ -166,56 +173,7 @@ Algorithm~\ref{algo:forwardeuler} provides an overview on how that can be achiev \end{algorithm} \end{figure} -\subsection{4th order Runge-Kutta} - -For a particle $i$, we can express the 4th order Runge-Kutta (RK4) method as -\begin{equation} - \begin{split} - \vb{v}_{i,j+1} &= \vb{v}_{i,j} + \frac{h}{6} \left( \vb{k}_{\vb{v},1,i} - + 2\vb{k}_{\vb{v},2,i} + 2\vb{k}_{\vb{v},3,i} + \vb{k}_{\vb{v},4,i} - \right) \\ - \vb{r}_{i,j+1} &= \vb{r}_{i,j} + \frac{h}{6} \left( \vb{k}_{\vb{r},1,i} - + 2\vb{k}_{\vb{r},2,i} + 2\vb{k}_{\vb{r},3,i} + \vb{k}_{\vb{r},4,i} - \right), - \end{split} -\end{equation} -where -\begin{equation} - \begin{split} - \vb{k}_{\vb{v},1,i} &= \frac{\vb{F}_i(t_j, \vb{v}_{i,j}, - \vb{r}_{i,j})}{m} \\ - \vb{k}_{\vb{r},1,i} &= \vb{v}_{i,j} \\ - \vb{k}_{\vb{v},2,i} &= \frac{\vb{F}_i(t_j+\frac{h}{2}, \vb{v}_{i,j} - + h \cdot \frac{\vb{k}_{\vb{v},1,i}}{2}, \vb{r}_{i,j} - + h \cdot \frac{\vb{k}_{\vb{r},1,i}}{2})}{m} \\ - \vb{k}_{\vb{r},2,i} &= \vb{v}_{i,j} - + h \cdot \frac{\vb{k}_{\vb{v},1,i}}{2} \\ - \vb{k}_{\vb{v},3,i} &= \frac{\vb{F}_i(t_j+\frac{h}{2}, \vb{v}_{i,j} - + h \cdot \frac{\vb{k}_{\vb{v},2,i}}{2}, \vb{r}_{i,j} - + h \frac{\cdot \vb{k}_{\vb{r},2,i}}{2})}{m} \\ - \vb{k}_{\vb{r},3,i} &= \vb{v}_{i,j} - + h \cdot \frac{\vb{k}_{\vb{v},2,i}}{2} \\ - \vb{k}_{\vb{v},4,i} &= \frac{\vb{F}_i(t_j+h, \vb{v}_{i,j} - + h \cdot \vb{k}_{\vb{v},3,i}, \vb{r}_{i,j} - + h \cdot \vb{k}_{\vb{r},3,i})}{m} \\ - \vb{k}_{\vb{r},4,i} &= \vb{v}_{i,j} - + h \cdot \frac{\vb{k}_{\vb{v},1,i}}{2}. - \end{split} -\end{equation} - -In order to find each $\vb{k}_{\vb{r},i}$ and $\vb{k}_{\vb{v},i}$, -we need to first compute all $\vb{k}_{\vb{r},i}$ and $\vb{k}_{\vb{v},i}$ -for all particles, then we can update the particles in order to compute -$\vb{k}_{\vb{r},i+1}$ and $\vb{k}_{\vb{v},i+1}$. In order for the algorithm -to work, we need to save a copy of each particle before starting so that we -can update the particles correctly for each step. - -This approach would require 8 loops to be able to complete the calculation since -we cannot update the particles until after all $\vb{k}$ values have been -computed, however if we create a temporary array that holds particles, we can -put the updated particles in there, and then use that array in the next loop, -and would reduce the required amount of loops down to 4. - +We also implemented the 4th order Runge-Kutta (RK4) method, using the expressin given in appendix \ref{sec:algo_rk4}. We define the RK4 algorithm in \ref{algo:rk4}. $\mathbf{F}$ does not take any arguments, however, the total force acting on the particle is calculated using the value of position and velocity within $particles$. \begin{figure} \begin{algorithm}[H] \caption{RK4 method} @@ -280,17 +238,20 @@ and would reduce the required amount of loops down to 4. \EndProcedure \end{algorithmic} \end{algorithm} - $\vb{F}$ in the algorithm does not take any arguments as it uses the - velocities and positions of the particles inside the array $particles$ to - calculate the total force acting on particle $i$. \end{figure} - - -\subsection{Testing the simulation} - -\subsection{Relative error and error convergance rate} +For a system of particles, the positional value of one particle affects the rest of the particles. We calculated the next step for all particles, updated their values in a copy, and updated the positional value based on the original values. 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_particle_cond}. + +\subsection{Testing and error analysis} +We implemented a test suite to validate our results, details can be found in appendix \ref{sec:testing}. To analyze our results, we compared the relative error of the analytical solution with the result from the implemented forward Euler method and RK4 method. We simulated four times using time steps given in table \ref{tab:time_steps}, and estimated the error convergence rate $r_{\text{err}}$ for both methods using +\begin{equation} + r_{\text{err}} = \frac{1}{3} \sum_{k=2}^{4} \frac{\log(\Delta_{\text{max},k} / \Delta_{\text{max}, k-1})}{\log(h_{k} / h_{k-1})}. +\end{equation} +The maximum error of a simulation $k$ with a step size $h_{k}$ is given by +\begin{equation*} + \Delta_{\text{max},k} = \max_{i} |\mathbf{r}_{i, \text{exact}} - \mathbf{r}_{i}|, +\end{equation*} +where $\mathbf{r}_{i, \text{exact}}$ is the analytical solution, and $\mathbf{r}_{i}$ is the computed result. \subsection{Tools} -The numerical methods implemented in C++, are parallelized using OpenMP \cite{openmp:2018}. We used the Python library matplotlib \cite{hunter:2007:matplotlib} to produce all the plots, and seaborn \cite{waskom:2021:seaborn} to set the theme in the figures. - -\end{document} +The numerical methods are implemented in C++, and parallelized using \verb|OpenMP| \cite{openmp:2018}. In addition, we used profiler Scalasca and score-p instrumentation We used the Python library \verb|matplotlib| \cite{hunter:2007:matplotlib} to produce all the plots, and \verb|seaborn| \cite{waskom:2021:seaborn} to set the theme in the figures. +\end{document} \ No newline at end of file diff --git a/latex/sections/results.tex b/latex/sections/results.tex index 2dfa504..2c7fe45 100644 --- a/latex/sections/results.tex +++ b/latex/sections/results.tex @@ -2,88 +2,108 @@ \graphicspath{{\subfix{../images/}}} \begin{document} - \section{Results and Discussion} -% Single particle -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}. +The simulations were performed using a constant configuration for the Penning trap, where the values of $B_{0}$, $V_{0}$, and $d$ can be found in table \ref{tab:penning_config}. We also used a constant configuration for the particles, found in table \ref{tab:particle_config}. Initial position of particle 1 $p_{1}$ was set to $(20, 0, 20) \mu m$ with velocity $(0, 25, 0) \mu m/ \mu s$, whereas the position of particle 2 $p_{2}$ was set to $(25, 25, 0) \mu m$ with velocity $(0, 40, 5) \mu m/ \mu s$. -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. -% Add something about why the simulated result is good, cos(wt) when w is -% -\begin{table}[H] - \centering - \begin{tabular}[c]{lll} - Particle & Position & Velocity \\ - \hline - $p_{1}$ & $(20, 0, 20) \mu m$ & $(0, 25, 0) \mu m/ \mu s$ \\ - $p_{2}$ & $(25, 25, 0) \mu m$ & $(0, 40, 5) \mu m/ \mu s$ \\ - \hline - \end{tabular} - \caption{Initial position and velocity of particle 1 ($p_{1}$) particle 2 ($p_{2}$), where the analytical solution is given by $z(t) = z_{0} \cos (\omega_{z} t)$} - \label{tab:initial_condition_particles} -\end{table} +First, we simulated a single particle for $50 \mu s$, approximating the particle's motion using the RK4 method. In addition we compared the result with the analytical solution in figure \ref{fig:single_particle}. The approximated solution completely overlap the analytical, suggesting the implemented method is approximating the solution with minimal error. The angular frequency is +\begin{align*} + \omega_{z} &= \sqrt{\frac{2 q V_{0}}{m d^{2}}} = \sqrt{\frac{2 \cdot 1 \cdot 9.65}{40.078}} \approx 0.694 \text{ rad/\textmu s}, +\end{align*} +which result in a period (P) of +\begin{align*} + \text{P} &= \frac{2 \pi}{|\omega_{z}|} \approx 9.054 \text{\textmu s}. +\end{align*} +From figure \ref{fig:single_particle} we see that the period, the time it takes for the particle to reach the same position, is close to 9 \textmu s. This is what we would expect given the value of $\omega_{z}$. +% Figure: single_particle.pdf \begin{figure}[H] \centering \includegraphics[width=\linewidth]{images/single_particle.pdf} - \caption{A single particle in the Penning trap, approximated and analytical motion in z-direction.} + \caption{Movement of a single particle in z-direction. Approximated using the 4th order Runge-Kutta method, compared to the analytical solution $z(t) = z_{0} \cos (\omega_{z} t)$.} \label{fig:single_particle} \end{figure} -% Add equations of motion for particle with interaction eq. (18, 19, 20) -% Multiple particles -% Add initial condition of Penning trap -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. -\begin{table}[H] + +To evaluate the implemented methods we simulated the particle using both forward Euler (FE) and RK4, with different time steps given in \ref{tab:time_steps}. Again, we simulated the particle movement for $50$ \textmu s, and estimated the relative error of each method. The result can be found in figure \ref{fig:relative_error}. The relative error of FE is large compared to the relative error of RK4. The error convergence rate for FE is $r_{\text{err}} \approx 1.39652$, and RK4 $r_{\text{err}} \approx 3.99998$, which means RK4 converges faster compared to FE. +\begin{figure}[H] \centering - \begin{tabular}{lll} - $B_0$ & $V_{0}$ & $d$ \\ - \hline - \end{tabular} - \caption{Caption} - \label{tab:initial_condition_penning} -\end{table} -% -\begin{figure} - \centering - \includegraphics[width=\linewidth]{images/plot_2_particles_xy.pdf} - \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.} - \label{fig:two_particles} + \includegraphics[width=\linewidth]{images/relative_error.pdf} + \caption{Relative error measured for 4th order Runge-Kutta \(a\) and foreward Euler \(b\). Important to notice when reading the plot, that the scale of measured relative error differ between the methods.} + \label{fig:relative_error} \end{figure} -% Phase space plot -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. -% -\begin{figure} +RK4 is a more complex method, which requires four times the calculations compared to FE. When the simulation is done with $n_{4} = 32000$ time steps, the difference in relative error decreases. The cost of calculation exceed the benefit of accuracy. That is, at a large number of time steps we could find a satisfying approximation to the solution using FE, with less calculations. + +Next, we simulated the movement of two particles in radial direction. In figure \ref{fig:two_particles_radial}, we see the path of both particles without interaction. The starting position of particle 1 $p_{1}$ is closer to the center of the Penning trap, than the starting position of $p_{}$. Both particles move in an orbital path arount the center, however, the starting velocity of each particles determine the minimum and maximum distance a particle moves from the center. We find these values bounds using eq. \ref{eq:upper_b} and eq. \ref{eq:lower_b} +\begin{align*} + R_{+} &= A_{+} + A_{-} \\ + &= -12.3232 \text{\textmu m} + 32.3232 \text{\textmu m} \\ + &= 20.00 \text{\textmu m}, +\end{align*} +and +\begin{align*} + R_{-} &= |A_{+} - A_{-}| \\ + &= |-12.3232 \text{\textmu m} - 32.3232 \text{\textmu m}| \\ + &= 44.6464 \text{\textmu m}. +\end{align*} +The distance is similar that in figure \ref{fig:two_particles_radial}. In addition, we can see from \ref{fig:phase_nointer_x} that the distance differ between $p_{1}$ and $p_{2}$. When we study the particle movement in z-direction, in figure \ref{fig:phase_nointer_z}, we see a circular movement. Again, the distance the particle moves from the center is determined by its initial conditions. + +In figure \ref{fig:two_particles_radial_interaction} we see the movement in radial direction, where particle interaction is included. The orbital path is similar to what we see without particle interaction. However, the additional force acting on the particle is affecting it's trajectory. The distance each particle is moving from the center, differs depending on the position. Which is also seen in figure \ref{fig:phase_inter_x}, where we see the trajectory of the particle at a given position. When we study the phase space plot, where particle interaction is included, in figure \ref{fig:phase_inter_z} a similar difference is observed. In figure \ref{fig:3d_particles} we see the movement of two particles in both the radial direction and z-direction. The figure include particle movement with and without interaction, where we can observe the change in movement.% Something about the trajectory related to the interaction? + +\begin{figure}[H] \centering - \includegraphics[width=\linewidth]{images/phase_space_no_interaction_x.pdf} - \caption{Phase space plot of two particles in x-direction.} - \label{fig:phase_space_2x} -\end{figure} -% -\begin{figure} - \centering - \includegraphics[width=\linewidth]{images/phase_space_interaction_z.pdf} - \caption{Phase space plot of two particles in z-direction.} - \label{fig:phase_space_2z} + \includegraphics[width=\linewidth]{images/simulate_2_particles_no_interaction_xy.pdf} + \caption{Path of two particles, $p_{1}$ and $p_{2}$, without interaction. The black dot marks the starting position of each particle.} + \label{fig:two_particles_radial} \end{figure} -\begin{figure} +\begin{figure}[H] + \centering + \includegraphics[width=\linewidth]{images/simulate_2_particles_interaction_xy.pdf} + \caption{Path of two particles, $p_{1}$ and $p_{2}$, with interaction. The black dot marks the starting position of each particle.} + \label{fig:two_particles_radial_interaction} +\end{figure} + +\begin{figure}[H] + \centering + \includegraphics[width=\linewidth]{images/phase_space_no_interaction_x.pdf} + \caption{Particle trajectory at a given position $x$, without particle interaction. The black dot marks the starting position of each particle.} + \label{fig:phase_nointer_x} +\end{figure} + +\begin{figure}[H] + \centering + \includegraphics[width=\linewidth]{images/phase_space_no_interaction_z.pdf} + \caption{Particle trajectory at a given position $z$, without particle interaction. The black dot marks the starting position of each particle.} + \label{fig:phase_nointer_z} +\end{figure} + +\begin{figure}[H] + \centering + \includegraphics[width=\linewidth]{images/phase_space_interaction_x.pdf} + \caption{Particle trajectory at a given position $x$, with particle interaction. The black dot marks the starting position of each particle.} + \label{fig:phase_inter_x} +\end{figure} + +\begin{figure}[H] + \centering + \includegraphics[width=\linewidth]{images/phase_space_interaction_z.pdf} + \caption{Particle trajectory at a given position $z$, with particle interaction. The black dot marks the starting position of each particle.} + \label{fig:phase_inter_z} +\end{figure} + +\begin{figure}[H] \centering \includegraphics[width=\linewidth]{images/3d_plot.pdf} - \caption{3D plot of particles-} + \caption{The movement of particles in a Penning trap. Where $p_{1}$ and $p_{2}$ is without interaction, $\hat{p}_{1}$ and $\hat{p}_{2}$ is without interaction.} \label{fig:3d_particles} \end{figure} -\begin{figure} - \centering - \includegraphics[width=\linewidth]{images/relative_error.pdf} - \caption{Relative error of RK4 and forward Euler method.} - \label{fig:rel_err} -\end{figure} +Finally, by subjecting the system to a time-dependent field, making the replacement in \ref{eq:pertubation}, we study the fraction of particles left at different amplitudes $f$. We can see how the different amplitudes lead to loss of particles, at different angular frequencies $\omega_{V}$ in \ref{fig:wide_sweep}. % Something - -\begin{figure} +When we study ... closer, in figure \ref{fig:} , we observe ... Certain angular frequencies are more effective in pushing particles out of the Penning trap. +\begin{figure}[H] \centering \includegraphics[width=\linewidth]{images/particles_left_wide_sweep.pdf} - \caption{Fraction of particles left in the Penning trap, with a given amplitude $f$.} - \label{fig:particles_left} + \caption{Exploring particles} + \label{fig:wide_sweep} \end{figure} -\end{document} + +\end{document} \ No newline at end of file -- 2.20.1 From 87d13194f34972b44f31e4ca317574bfeb88b57c Mon Sep 17 00:00:00 2001 From: Janita Willumsen Date: Tue, 24 Oct 2023 17:21:53 +0200 Subject: [PATCH 03/23] Update references --- latex/references/references.bib | 13 +++++++++++-- 1 file changed, 11 insertions(+), 2 deletions(-) diff --git a/latex/references/references.bib b/latex/references/references.bib index 341ff73..1f37d96 100644 --- a/latex/references/references.bib +++ b/latex/references/references.bib @@ -97,8 +97,17 @@ urldate = {2023-10-22} } -@misc{penning:fig, - key = {something}, +@online{penning:fig, + author = {Ludwig-Maximilians-Universität München}, + title = {Penning traps}, url = {https://www.med.physik.uni-muenchen.de/research/nuclear-science/nuclear-masses/mlltrap/layout/traps/index.html}, + urldate = {2023-10-23} note = {Configuration figure a} } + +@online{github:repo, + author = {Cory Alexander Balaton and Janita Ovidie Sandtrøen Willumsen}, + url = {https://github.uio.no/FYS3150-G2-2023/Project-3}, + urldate = {2023-10-24}, + note = {Complete repo of project 3} +} \ No newline at end of file -- 2.20.1 From bd7f911ec59b27ff4a22af8b3c08cb6a8552c4b5 Mon Sep 17 00:00:00 2001 From: Janita Willumsen Date: Tue, 24 Oct 2023 17:22:18 +0200 Subject: [PATCH 04/23] Move appendix --- latex/appendix.tex | 266 +++++++++++++++++++++++++++++++++++++++++++++ 1 file changed, 266 insertions(+) create mode 100644 latex/appendix.tex diff --git a/latex/appendix.tex b/latex/appendix.tex new file mode 100644 index 0000000..dabf25e --- /dev/null +++ b/latex/appendix.tex @@ -0,0 +1,266 @@ +\documentclass[../main.tex]{subfiles} +\graphicspath{{\subfix{../images/}}} + +\begin{document} +\section{Derivation of equations}\label{sec:derivations} +% Problem 1 +\subsection{Equations of motion}\label{sec:eq_motion} % +First, we need to define the velocity of the particle +\begin{align*} + \mathbf{v} \equiv \frac{d \mathbf{r}}{dt} &= \bigg( \frac{dx}{dt}, \frac{dy}{dt}, \frac{dz}{dt} \bigg). +\end{align*} % +% +We can rewrite the velocity as $\dot{r} = (\dot{x}, \dot{y}, \dot{z})$, and find the cross product +\begin{align*} + q \mathbf{v} \cross \mathbf{B} &= + q \begin{vmatrix} + \hat{e}_{x} & \hat{e}_{y} & \hat{e}_{z} \\ + \dot{x} & \dot{y} & \dot{z} \\ + 0 & 0 & B_{0} + \end{vmatrix} + = q \big( B_{0} \dot{y}, -B_{0} \dot{x}, 0 \big). +\end{align*} % +% +We are considering an ideal Penning traps, where we define the electric potential as +\begin{align*} + V(x, y, z) &= \frac{V_{0}}{2 d^{2}}(2z^{2} - x^{2} - y^{2}). +\end{align*} % +% +The relationship between the electric field $\mathbf{E}$ and the electric potential of the field is given by +\begin{align*} + \mathbf{E} &= - \nabla V \\ + &= - \bigg( \frac{dV}{dx}, \frac{dV}{dy} \frac{dV}{dz} \bigg) \\ + &= \frac{V_{0}}{d^{2}} \big( x, y, -2z \big). +\end{align*} % +% +We can now express the Lorentz force as +\begin{align*} + \mathbf{F} &= q \mathbf{E} + q \mathbf{v} \cross \mathbf{B} \\ + &= \frac{q V_{0}}{d^{2}} \big( x, y, -2z \big) + \big(q B_{0} \dot{y}, -q B_{0} \dot{x}, 0 \big), +\end{align*} % +% +and insert it into Newtons equation \eqref{eq:newton_second}. We get +\begin{align*} + \ddot{\mathbf{r}} &= \bigg( \frac{q V_{0}}{m d^{2}} x, \frac{q V_{0}}{m d^{2}} y, -\frac{2 q V_{0}}{m d^{2}} z \bigg) + \bigg(\frac{q B_{0}}{m} \dot{y}, -\frac{q B_{0}}{m} \dot{x}, 0 \bigg), +\end{align*} % +% +which can be written as +\begin{align*} + \ddot{x} &= \frac{q V_{0}}{m d^{2}} x + \frac{q B_{0}}{m} \dot{y}, \\ + \ddot{y} &= \frac{q V_{0}}{m d^{2}} y - \frac{q B_{0}}{m} \dot{x}, \\ + \ddot{z} &= -\frac{2 q V_{0}}{m d^{2}} z. +\end{align*} % +% +If we define +\begin{equation*} + \omega_{0} \equiv \frac{q B_{0}}{m}, \quad \omega_{z}^{2} \equiv \frac{2 q V_{0}}{m d^{2}}, +\end{equation*} % +% +the equations of motion can be written as +\begin{align*} + \ddot{x} &= \frac{1}{2} \omega_{z}^{2} x + \omega_{0} \dot{y}, \\ + \ddot{y} &= \frac{1}{2} \omega_{z}^{2} y - \omega_{0} \dot{x}, \\ + \ddot{z} &= -\omega_{z}^{2} z. \\ +\end{align*} % +% +\subsection{General solution}\label{sec:eq_general} +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}}. +\end{align*} % +The characteristic equation has two complex roots +\begin{equation*} + r_{1} = - i \omega_{z}, \quad r_{2} = i \omega_{z}, +\end{equation*} +which give us solutions in the general form +\begin{equation*} + z = c_{1} e^{i \omega_{z} t} + c_{2} e^{-i \omega_{z} t}. +\end{equation*} +In addition, 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 the general solution 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}) \\ + & \quad + c_{2} (\cos{\omega_{z} t} - i \sin{\omega_{z} t} \\ + &= E \cos{\omega_{z} t} + i F \sin{\omega_{z} t}. +\end{align*} % +% +\subsection{Complex function}\label{sec:eq_complex} % +In sec. \ref{sec:eq_motion} we found the differential equations for $\ddot{x}$ and $\ddot{y}$. To derive a single differential equation, we introduce the complex function $f(t) = x(t) + iy(t)$, which gives us +\begin{align*} + 0 &= \Big(\ddot{x} - \omega_{0} \dot{y} - \frac{1}{2} \omega_{z}^{2} x \Big) + i \Big(\ddot{y} + \omega_{0} \dot{x} - \frac{1}{2} \omega_{z}^{2} y \Big) \\ + &= \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 \\ + &= \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. \\ +\end{align*} +Using the definition $i = \sqrt{-1}$, we can rewrite +\begin{equation*} + i \omega_{0} \dot{x} + (-1) \omega_{0} \dot{y} = i \omega_{0} \dot{x} + i^{2} \omega_{0} \dot{y}. +\end{equation*} +This gives us a single differential equation +\begin{align*} + 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 \\ + &= \ddot{f} + i \omega_{0} \dot{f} - \frac{1}{2} \omega_{z}^{2} f. +\end{align*} +% +\subsection{Physical coordinates}\label{sec:eq_coord} +We can rewrite eq. \eqref{eq:general_solution}, as +\begin{align*} + f(t) &= A_{+}e^{-i(\omega_{+} t + \phi_{+})} + A_{-}e^{-i(\omega_{-} t + \phi_{-})} \\ + &= A_{+}(\cos{(\omega_{+} t + \phi_{+})} - i \sin{(\omega_{+} t + \phi_{+})}) \\ + % \numberthis \label{eq:general_solution_trig} + & \quad + A_{-}(\cos{(\omega_{-} t + \phi_{-})} - i \sin{(\omega_{-} t + \phi_{-})}). +\end{align*} % +% +\subsection{Upper and lower bounds}\label{sec:upper_lower_bound} +To obtain the upper and lower bounds of the particle's distance from the origin, we first find an expression for the second norm defined as $|f(t)| = \sqrt{(x(t))^{2} + (y(t))^{2}}$. +\begin{align*} + (x(t))^{2} &= \big( A_{+}\cos(\omega_{+} t + \phi_{+}) + A_{-}\cos(\omega_{-} t + \phi_{-}) \big)^{2} \\ + &= A_{+}^{2} \cos^{2}(\omega_{+} t + \phi_{+}) \\ + & \quad + 2 A_{+}A_{-} \cos(\omega_{+} t + \phi_{+})\cos(\omega_{-} t + \phi_{-}) \\ + & \quad + A_{-}^{2}\cos^{2}(\omega_{-} t + \phi_{-}), \\ +\end{align*} % +\begin{align*} + (y(t))^{2} &= \big( - A_{+} \sin(\omega_{+} t + \phi_{+}) - A_{-} \sin(\omega_{-} t + \phi_{-}) \big)^{2} \\ + &= A_{+}^{2} \sin^{2}(\omega_{+} t + \phi_{+}) \\ + & \quad + 2 A_{+}A_{-} \sin(\omega_{+} t + \phi_{+})\sin(\omega_{-} t + \phi_{-}) \\ + & \quad + A_{-}^{2} \sin^{2}(\omega_{-} t + \phi_{-}). +\end{align*} % +We insert these expressions, and find +\begin{align*} + |f(t)| &= \sqrt{(x(t))^{2} + (y(t))^{2}} \\ + &= \sqrt{A_{+}^{2} + 2 A_{+} A_{-} \cos^{2}(\alpha) + A_{-}^{2}}, +\end{align*} % +where $\alpha = (\omega_{+} - \omega_{-}) t +( \phi_{+} - \phi_{-})$. If we set $\alpha = 0$ we get $\cos(0) = 1$, and obtain the upper bound +\begin{align*} + R_{+} &= \sqrt{A_{+}^{2} + 2 A_{+} A_{-} + A_{-}^{2}} \\ + &= \sqrt{(A_{+} + A_{-})^{2}} \\ + &= A_{+} + A_{-}. +\end{align*} +If $\alpha = \pi$ we get $\cos(\pi) = -1$, and find the lower bound +\begin{align*} + R_{-} &= \sqrt{A_{+}^{2} - 2 A_{+} A_{-} + A_{-}^{2}} \\ + &= \sqrt{(A_{+} - A_{-})^{2}} \\ + &= |A_{+} - A_{-}|. +\end{align*} % +% +\subsection{Bounded solution}\label{sec:bounded_solution} +To find a bounded solution, we need to consider the angular rate in eq. \eqref{eq:angular_rate}. Specifically the square-root expression +\begin{align*} + \sqrt{\omega_{0}^{2} - 2 \omega_{z}^{2}}. +\end{align*} +A bounded solution can be found when this expression is greater than zero. Meaning +\begin{align*} + \omega_{0}^{2} &> 2 \omega_{z}^{2}. +\end{align*} % +% +We can now use the definition of $\omega_{0}$ and $\omega_{z}^{2}$ to find an expression related to the Penning trap parameters, and the particle properties, to get +\begin{align*} + \Big( \frac{q B_{0}}{m} \Big)^{2} &> 2 \frac{2 q V_{0}}{m d^{2}} \\ + \frac{q^{2} B_{0}^{2}}{m^{2}} &> \frac{4 q V_{0}}{m d^{2}}. +\end{align*} % + +\section{Values used in simulation} +\subsection{Specific analytical solution}\label{sec:spec_analytical} +To compare our implementation of the Penning trap and particle, we have to specify initial conditions of the system to compare with the analytical solution. The initial conditions of a particle with a single positive charge, can be found in table \ref{tab:initial_particle_cond}. +\begin{table}[H] + \centering + \begin{tabular}[c]{ll} + $\mathbf{r}(0)$ & $\dot{\mathbf{r}}(0)$ \\ + \hline + $x(0)$ = $x_{0}$ & $\dot{x}(0)$ = $0$ \\ + $y(0)$ = $0$ & $\dot{y}(0)$ = $v_{0}$ \\ + $z(0)$ = $z_{0}$ & $\dot{z}(0)$ = $0$ \\ + \hline + \end{tabular} + \caption{Initial values of a particle with a single charge $q$, and mass $m$, confined in a Penning trap.} + \label{tab:initial_particle_cond} +\end{table} + +\begin{table}[H] + \centering + \begin{tabular}[c]{lll} + Property & Value \\ + \hline + $B_{0}$ & $1.00 T$ = $9.65 \cross 10^{1} \frac{u}{(\mu s) e}$ \\ + $V_{0}$ & $25.0 mV$ = $2.41 \cross 10^{6} \frac{u (\mu m)^{2}}{(\mu s)^{2} e}$ \\ + $d$ & $500 \ \mu m$ = \\ + \hline + \end{tabular} + \caption{Default configuration of the Penning trap, where the value of T and V can be found in table \ref{tab:constants}.} + \label{tab:penning_config} +\end{table} + +\begin{table}[H] + \centering + \begin{tabular}[c]{lll} + Property & Value \\ + \hline + $q$ & $1.00 T$ & $9.65 \cross 10^{1} \frac{u}{(\mu s) e}$ \\ + $m$ & $25.0 mV$ & $2.41 \cross 10^{6} \frac{u (\mu m)^{2}}{(\mu s)^{2} e}$ \\ + \hline + \end{tabular} + \caption{Default configuration of the Penning trap, where the value of T and V can be found in table \ref{tab:constants}.} + \label{tab:particle_config} +\end{table} + +\begin{table}[H] + \centering + \begin{tabular}[c]{lll} + $n_{k}$ & Time steps & Step size \\ + \hline + $n_{1}$ & $4000$ & $0.0125$ \\ + $n_{2}$ & $8000$ & $0.00625$ \\ + $n_{3}$ & $16000$ & $0.003125$ \\ + $n_{4}$ & $32000$ & $0.0015625$ \\ + \hline + \end{tabular} + \caption{Number of steps used in a given simulation $k$, where the step size corresponds to $h_{1} = 50 / n_{k} \mu s$.} + \label{tab:time_steps} +\end{table} + +\subsection{Numbers and units}\label{sec:numbers_units} +\begin{table}[H] + \centering + \begin{tabular}[c]{lll} + Constant & Value & Unit \\ + \hline + $k_{e}$ (Coulomb) & $1.38935333 \cross 10^{5}$ & $\frac{u (\mu m)^{3}}{(\mu s)^{2} e^{2}}$ \\ + T (Tesla) & $9.64852558 \cross 10^{1}$ & $\frac{u}{(\mu s) e}$ \\ + V (Volt) & $9.64852558 \cross 10^{7}$ & $\frac{u (\mu m)^{2}}{(\mu s)^{2} e}$ \\ + \hline + \end{tabular} + \caption{Value of the Coulomb constant ($k_{e}$), and the SI units for magnetic field strength ($T$) and electric potential ($V$). The base units are given by length in micrometre ($\mu m$), time in microseconds ($\mu s$), mass in ($u$), and charge in elementary charge ($e$).} + \label{tab:constants} +\end{table} + +\section{Algorithm implementation and testing}\label{sec:algo} +\subsection{Forward Euler}\label{sec:algo_euler} +For a particle $i$, at time step $j$, the forward Euler method for a coupled system can be expressed as +\begin{align*} + \mathbf{r}_{i,j+1} &= \mathbf{r}_{i,j} + h \frac{d \mathbf{r}_{i,j}}{dt} = \mathbf{r}_{i,j} + h \mathbf{v}_{i,j} \\ + \mathbf{v}_{i,j+1} &= \mathbf{v}_{i,j} + h \frac{\mathbf{v}_{i,j}}{dt} = \mathbf{v}_{i,j} + h \frac{\mathbf{F}(t_{j},\mathbf{v}_{i,j}, \mathbf{r}_{i,j} )}{m_{i}}, +\end{align*} +$m_{i}$ is the mass of the particle, and $h$ is the step length. + +\subsection{4th order Runge-Kutta}\label{sec:algo_rk4} +For a particle $i$, at time step $j$, the 4th order Runge-Kutta method for a coupled system can be expressed as +\begin{align*} + \mathbf{v}_{i,j+1} &= \mathbf{v}_{i,j} + \frac{h}{6} (\mathbf{k}_{\mathbf{v},1,i} + 2 \mathbf{k}_{\mathbf{v},2,i} + 2 \mathbf{k}_{\mathbf{v},3,i} + \mathbf{k}_{\mathbf{v},4,i} ), \\ + \mathbf{r}_{i,j+1} &= \mathbf{r}_{i,j} + \frac{h}{6} (\mathbf{k}_{\mathbf{r},1,i} + 2 \mathbf{k}_{\mathbf{r},2,i} + 2\mathbf{k}_{\mathbf{r},3,i} + \mathbf{k}_{\mathbf{r},4,i}), +\end{align*} +where +\begin{align*} + \mathbf{k}_{\mathbf{r},1,i} &= \mathbf{v}_{i,j} \\ + \mathbf{k}_{\mathbf{v},1,i} &= \frac{\mathbf{F}_{i}(t_{j}, \mathbf{v}_{i,j}, \mathbf{r}_{i,j})}{m_{i}}, \\ + \mathbf{k}_{\mathbf{r},2,i} &= \mathbf{v}_{i,j} + h \frac{\mathbf{k}_{\mathbf{v},1,i}}{2}, \\ + \mathbf{k}_{\mathbf{v},2,i} &= \frac{\mathbf{F}_{i}(t_{j}+\frac{h}{2}, \mathbf{v}_{i,j} + h \frac{\mathbf{k}_{\mathbf{v},1,i}}{2}, \mathbf{r}_{i,j} + h \frac{\mathbf{k}_{\mathbf{r},1,i}}{2})}{m_{i}}, \\ + \mathbf{k}_{\mathbf{r},3,i} &= \mathbf{v}_{i,j} + h \frac{\mathbf{k}_{\mathbf{v},2,i}}{2}, \\ + \mathbf{k}_{\mathbf{v},3,i} &= \frac{\mathbf{F}_{i}(t_{j}+\frac{h}{2}, \mathbf{v}_{i,j} + h \frac{\mathbf{k}_{\mathbf{v},2,i}}{2}, \mathbf{r}_{i,j} + h \frac{\mathbf{k}_{\mathbf{r},2,i}}{2})}{m_{i}}, \\ + \mathbf{k}_{\mathbf{r},4,i} &= \mathbf{v}_{i,j} + h \frac{\mathbf{k}_{\mathbf{v},1,i}}{2}, \\ + \mathbf{k}_{\mathbf{v},4,i} &= \frac{\mathbf{F}_{i}(t_{j}+h, \mathbf{v}_{i,j} + h \mathbf{k}_{\mathbf{v},3,i}, \mathbf{r}_{i,j} + h \mathbf{k}_{\mathbf{r},3,i})}{m_{i}}. +\end{align*} +In order to find each $\mathbf{k}_{\mathbf{r},i}$ and $\mathbf{k}_{\mathbf{v},i}$, we need to first compute all $\mathbf{k}_{\mathbf{r},i}$ and $\mathbf{k}_{\mathbf{v},i}$ for all particles, then update the particle values in order to compute $\mathbf{k}_{\mathbf{r},i+1}$ and $\mathbf{k}_{\mathbf{v},i+1}$. + +\subsection{Test of implementation}\label{sec:testing} +We implemented a test suite, to validate the implementation during code development. In addition, we have implemented functionality to get informative output while testing the code. Further instructions with code can be found in the github repo \cite{github:repo}. + +\end{document} \ No newline at end of file -- 2.20.1 From 9e01ad8351ac408393d24e208ebf4c877b420175 Mon Sep 17 00:00:00 2001 From: Janita Willumsen Date: Tue, 24 Oct 2023 17:26:02 +0200 Subject: [PATCH 05/23] Add from overleaf draft --- latex/main.blg | 53 ------------------------------------------------- latex/main.pdf | Bin 637768 -> 734326 bytes latex/main.tex | 10 ++++++---- 3 files changed, 6 insertions(+), 57 deletions(-) delete mode 100644 latex/main.blg diff --git a/latex/main.blg b/latex/main.blg deleted file mode 100644 index 23ebaca..0000000 --- a/latex/main.blg +++ /dev/null @@ -1,53 +0,0 @@ -This is BibTeX, Version 0.99d (TeX Live 2022/CVE-2023-32700 patched) -Capacity: max_strings=200000, hash_size=200000, hash_prime=170003 -The top-level auxiliary file: main.aux -The style file: unsrt.bst -Database file #1: mainNotes.bib -Database file #2: references/references.bib -Warning--entry type 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index e7ab021..f73c54a 100644 --- a/latex/main.tex +++ b/latex/main.tex @@ -49,7 +49,7 @@ \begin{document} -\title{Project 3} % self-explanatory +\title{Simulating Particles in an Ideal Penning Trap} \author{Cory Alexander Balaton \& Janita Ovidie Sandtrøen Willumsen \\ \faGithub \, \url{https://github.uio.no/FYS3150-G2-2023/Project-3}} % self-explanatory \date{\today} % self-explanatory \noaffiliation % ignore this, but keep it. @@ -73,11 +73,13 @@ % Conclusion \subfile{sections/conclusion} +\newpage \appendix -\subfile{appendix/appendix_a} -\subfile{appendix/appendix_b} -\subfile{appendix/appendix_c} +\subfile{appendix} +% \subfile{appendix/appendix_b} +% \subfile{appendix/appendix_c} +% \subfile{appendix/appendix_d} \onecolumngrid -- 2.20.1 From 6f34b25e5c72de15842d7734726d8fd733265236 Mon Sep 17 00:00:00 2001 From: Cory Date: Tue, 24 Oct 2023 21:02:42 +0200 Subject: [PATCH 06/23] Fixed bug --- latex/references/references.bib | 4 ++-- 1 file changed, 2 insertions(+), 2 deletions(-) diff --git a/latex/references/references.bib b/latex/references/references.bib index 1f37d96..d106fc7 100644 --- a/latex/references/references.bib +++ b/latex/references/references.bib @@ -101,7 +101,7 @@ author = {Ludwig-Maximilians-Universität München}, title = {Penning traps}, url = {https://www.med.physik.uni-muenchen.de/research/nuclear-science/nuclear-masses/mlltrap/layout/traps/index.html}, - urldate = {2023-10-23} + urldate = {2023-10-23}, note = {Configuration figure a} } @@ -110,4 +110,4 @@ url = {https://github.uio.no/FYS3150-G2-2023/Project-3}, urldate = {2023-10-24}, note = {Complete repo of project 3} -} \ No newline at end of file +} -- 2.20.1 From 2b83889039f7663443c9a2fad2f0cc6f2b5747f6 Mon Sep 17 00:00:00 2001 From: Cory Date: Tue, 24 Oct 2023 21:03:15 +0200 Subject: [PATCH 07/23] Made some changes --- latex/sections/methods.tex | 18 +++++++++--------- latex/sections/results.tex | 16 ++++++++++++++-- 2 files changed, 23 insertions(+), 11 deletions(-) diff --git a/latex/sections/methods.tex b/latex/sections/methods.tex index aaace5a..2cb22ae 100644 --- a/latex/sections/methods.tex +++ b/latex/sections/methods.tex @@ -33,7 +33,7 @@ First, we consider Newton's second law \eqref{eq:newton_second}, to determine th m \ddot{\mathbf{r}} = \sum_{i} \mathbf{F}_{i} \end{equation} % % -In addition, we introduce the Lorentz force \eqref{eq:lorentz_force}, which describes the force acting the particle. +In addition, we introduce the Lorentz force \eqref{eq:lorentz_force}, which describes the force acting on the particle. \begin{equation}\label{eq:lorentz_force} \mathbf{F} = q \mathbf{E} + q \mathbf{v} \times \mathbf{B}, \end{equation} % @@ -122,7 +122,7 @@ We will also consider the numerical simulation of multiple particles, confined i \end{equation} % The Coulomb constant $k_{e}$ value can be found in table \ref{tab:constants}. -For multiple particles we have to modify the equations of motion, by adding a term for the force a given particle experience at a given point. When we scale eq. \eqref{eq:electric_field_interaction} by charge an mass, we get a new set of equations of motion +For multiple particles we have to modify the equations of motion, by adding a term for the force a given particle experience at a given point. When we scale eq. \eqref{eq:electric_field_interaction} by charge and mass, we get a new set of equations of motion \begin{align} \label{eq:coulomb_motion_x} \ddot{x}_{i} - \omega_{0,i} \dot{y}_{i} - \frac{1}{2} \omega_{z,i}^{2} x_{i} - k_{e} \frac{q_{i}}{m_{i}}\sum_{j \neq i} q_j \frac{x_{i} - x_{j}}{|\mathbf{r}_{i} - \mathbf{r}_{j}|^{3}} &= 0, \\ @@ -173,7 +173,7 @@ We first implemented the forward Euler method, using the expression for a couple \end{algorithm} \end{figure} -We also implemented the 4th order Runge-Kutta (RK4) method, using the expressin given in appendix \ref{sec:algo_rk4}. We define the RK4 algorithm in \ref{algo:rk4}. $\mathbf{F}$ does not take any arguments, however, the total force acting on the particle is calculated using the value of position and velocity within $particles$. +We also implemented the 4th order Runge-Kutta (RK4) method, using the expression given in appendix \ref{sec:algo_rk4}. We define the RK4 algorithm in \ref{algo:rk4}. $\mathbf{F}$ does not take any arguments, however, the total force acting on the particle is calculated using the value of position and velocity within $particles$. \begin{figure} \begin{algorithm}[H] \caption{RK4 method} @@ -225,12 +225,12 @@ We also implemented the 4th order Runge-Kutta (RK4) method, using the expressin \State $\vb{k}_{\vb{v},4,i} \leftarrow \frac{\vb{F}}{m}$ \State $tmp\_p_i.\vb{r} \leftarrow orig\_p_i.\vb{r} + \frac{dt}{6} - \cdot \left( \vb{k}_{\vb{r},1,i} + \vb{k}_{\vb{r},2,i} - + \vb{k}_{\vb{r},3,i} + \vb{k}_{\vb{r},4,i} \right)$ + \cdot \left( \vb{k}_{\vb{r},1,i} + 2 \vb{k}_{\vb{r},2,i} + + 2 \vb{k}_{\vb{r},3,i} + \vb{k}_{\vb{r},4,i} \right)$ \State $tmp\_p_i.\vb{v} \leftarrow orig\_p_i.\vb{v} + \frac{dt}{6} - \cdot \left( \vb{k}_{\vb{v},1,i} + \vb{k}_{\vb{v},2,i} - + \vb{k}_{\vb{v},3,i} + \vb{k}_{\vb{v},4,i} \right)$ + \cdot \left( \vb{k}_{\vb{v},1,i} + 2 \vb{k}_{\vb{v},2,i} + + 2 \vb{k}_{\vb{v},3,i} + \vb{k}_{\vb{v},4,i} \right)$ \EndFor @@ -253,5 +253,5 @@ The maximum error of a simulation $k$ with a step size $h_{k}$ is given by where $\mathbf{r}_{i, \text{exact}}$ is the analytical solution, and $\mathbf{r}_{i}$ is the computed result. \subsection{Tools} -The numerical methods are implemented in C++, and parallelized using \verb|OpenMP| \cite{openmp:2018}. In addition, we used profiler Scalasca and score-p instrumentation We used the Python library \verb|matplotlib| \cite{hunter:2007:matplotlib} to produce all the plots, and \verb|seaborn| \cite{waskom:2021:seaborn} to set the theme in the figures. -\end{document} \ No newline at end of file +The numerical methods are implemented in C++, and parallelized using \verb|OpenMP| \cite{openmp:2018}. In addition, we used profiler \verb|Scalasca| and \verb|score-p| instrumentation. We used the Python library \verb|matplotlib| \cite{hunter:2007:matplotlib} to produce all the plots, and \verb|seaborn| \cite{waskom:2021:seaborn} to set the theme in the figures. +\end{document} diff --git a/latex/sections/results.tex b/latex/sections/results.tex index 2c7fe45..9f5da99 100644 --- a/latex/sections/results.tex +++ b/latex/sections/results.tex @@ -98,7 +98,7 @@ In figure \ref{fig:two_particles_radial_interaction} we see the movement in radi Finally, by subjecting the system to a time-dependent field, making the replacement in \ref{eq:pertubation}, we study the fraction of particles left at different amplitudes $f$. We can see how the different amplitudes lead to loss of particles, at different angular frequencies $\omega_{V}$ in \ref{fig:wide_sweep}. % Something -When we study ... closer, in figure \ref{fig:} , we observe ... Certain angular frequencies are more effective in pushing particles out of the Penning trap. +When we study the angular frequency $\omega_{V} \in [1.1, 1.7]$ closer, in figure \ref{fig:narrow_sweep}, we observe that there are a few spots where more particles will escape. The most prominent one is where $\omega_V \in [1.1, 1.7]$, and when looking closer to the range, it seems like there's a resonating frequency at around $1.4MHz$ where All the particles will escape no matter the amplitude. When looking at the different angular frequencies with particle interaction like in \ref{fig:narrow_sweep_interactions}, we see that the amount of particles left is roughly the same as when there are no particle interactions, but that it's less predictable. \begin{figure}[H] \centering \includegraphics[width=\linewidth]{images/particles_left_wide_sweep.pdf} @@ -106,4 +106,16 @@ When we study ... closer, in figure \ref{fig:} , we observe ... Certain angular \label{fig:wide_sweep} \end{figure} -\end{document} \ No newline at end of file +\begin{figure}[H] + \centering + \includegraphics[width=\linewidth]{images/particles_left_narrow_sweep.pdf} + \caption{Exploring different angular frequencies more closely} + \label{fig:narrow_sweep} +\end{figure} +\begin{figure}[H] + \centering + \includegraphics[width=\linewidth]{images/particles_left_narrow_sweep_interactions.pdf} + \caption{Exploring particles} + \label{fig:wide_sweep_interactions} +\end{figure} +\end{document} -- 2.20.1 From b08d6babf018d06b27ddc5a3b824eee1336f96ea Mon Sep 17 00:00:00 2001 From: Cory Date: Tue, 24 Oct 2023 21:03:34 +0200 Subject: [PATCH 08/23] Update gitignore --- .gitignore | 1 + 1 file changed, 1 insertion(+) diff --git a/.gitignore b/.gitignore index b3816d1..2cdfb30 100644 --- a/.gitignore +++ b/.gitignore @@ -37,6 +37,7 @@ *.out *.synctex.gz *.bbl +*.blg latex/mainNotes.bib -- 2.20.1 From 4e09128483a48b0dcaa466277903e104008e8eb2 Mon Sep 17 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zYU9%j_vrFf7H2&(<%MUrYS-&f!`-|t=lj9G>1_j%w%D1`rkzYe{Ed+d^7o@YH1tIJ za9f;<;K{7n)Xcf!5ue5r=_sr~@1M4$l{SB5S69-G?Xd|vgEsdN#GJ$L?dhgBl-SauRw!l8?aefyfk(E?957NGWsxz@>_8N+VYVa30`gy8O zE@2wzvuu%S?~qA#zGy6DrI9M3d!{oJ?6l-&5GGo?nTusSExvdUR7>z^l`?CK)9aFa zWlEE}fbdz;&?iCsL^hWkbn+a#^>aahA$UKB&W^QqRQX3EK|7NXvA9(g*8BIVntdoI z-Vh_gCUSeoq_TSIxguc&DoGGhr~E5r*8V|)=_W)AyAsDh=^gnqNcTMRU|U)?+vsmB)J5T)fh}r?-584gm^2{V;uK6jz&FVP!T`n9^Qmz}T^u2r!Gp`xlXO3>MB*NsLciCw4ZK7#z-z^zbMQbxHa z9bCeX`v07pzWo)8S3awhKP)=9VP2Q7Xp`4p9%40O3dDCtuCaZ%sJ(rg_%m3)YDoF4 zw@_Dfu=DzTvdNzcA@0P*0?F`Io=hu~6yIGwy;POX4&k=6=x4gA8|E&J zw?q`=xuneM1?wUnv zyk!rfq$r&2ls>!1q6m|UB7CZvV0jHBE1X3>#^I1=|0~;nER(Rt!O>_!c)s6pa0h5! hB8UFpAE`U$M25%fm)15wVFX;1pC1T;$Z5y}{|8t!da?ij -- 2.20.1 From f06bfc0f59be212b6488a0d7225d789f7d378919 Mon Sep 17 00:00:00 2001 From: Cory Date: Tue, 24 Oct 2023 21:09:50 +0200 Subject: [PATCH 10/23] Add part about resonance --- latex/sections/conclusion.tex | 7 +++++-- 1 file changed, 5 insertions(+), 2 deletions(-) diff --git a/latex/sections/conclusion.tex b/latex/sections/conclusion.tex index 9a72d93..d0f5a1d 100644 --- a/latex/sections/conclusion.tex +++ b/latex/sections/conclusion.tex @@ -7,5 +7,8 @@ We studied the movement of particles confined by an ideal Penning trap, where we We also compared iterative methods with the analytical solution, and found that the forward Euler \(FE\) method result in an approximation with larger relative error than the 4th order Runge-kutta \(RK4\) method. In addition to a small relative error, we also found that RK4 has a higher convergence rate at approx. $4.0$, compared to FE at approx. $1.4$. Which suggest RK4 ... -Resonance and freq? -\end{document} \ No newline at end of file +We finally looked at how the amount of particles that stay in the Penning trap +change when using different angular frequencies in a time-dependent field. We +observed that there seem to be a resonating frequency at around $1.4MHz$ that +makes the particles escape the Penning trap even at low amplitudes. +\end{document} -- 2.20.1 From 978616f7d94f85ad277dcb612a31664f70d76258 Mon Sep 17 00:00:00 2001 From: Cory Date: Tue, 24 Oct 2023 21:12:45 +0200 Subject: [PATCH 11/23] Add resonance part for the abstract --- latex/sections/abstract.tex | 4 ++-- 1 file changed, 2 insertions(+), 2 deletions(-) diff --git a/latex/sections/abstract.tex b/latex/sections/abstract.tex index cb358a8..3632e5e 100644 --- a/latex/sections/abstract.tex +++ b/latex/sections/abstract.tex @@ -4,9 +4,9 @@ \begin{document} \begin{abstract} - We have studied the motion of singly-charged Calcium ions ($\text{Ca}^{+}$), inside an ideal Penning trap. With a numerical approach, studied the equations of motion by implementing the forward Euler method \(FE\) and the 4th order Runge-Kutta \(RK4\). We found that RK4 approximates the solution with smaller relative error than FE. In addition, we evaluated methods by rate of convergence. We found that RK4 has a higher convergence rate at approx. $4.0$, compared to FE at approx. $1.4$. + We have studied the motion of singly-charged Calcium ions ($\text{Ca}^{+}$), inside an ideal Penning trap. With a numerical approach, studied the equations of motion by implementing the forward Euler method \(FE\) and the 4th order Runge-Kutta \(RK4\). We found that RK4 approximates the solution with smaller relative error than FE. In addition, we evaluated methods by rate of convergence. We found that RK4 has a higher convergence rate at approx. $4.0$, compared to FE at approx. $1.4$. Finally, we observed that for a time-dependent field, that there is an angular frequency that resonates with the particles in such a way that they escape the Penning trap. Freq. \end{abstract} -\end{document} \ No newline at end of file +\end{document} -- 2.20.1 From baf780dfbb145829b52beb5a567871e1f0d33a83 Mon Sep 17 00:00:00 2001 From: Janita Willumsen Date: Tue, 24 Oct 2023 21:44:56 +0200 Subject: [PATCH 12/23] Finish report --- latex/appendix.tex | 10 +++++----- latex/main.pdf | Bin 750205 -> 737452 bytes latex/references/references.bib | 16 +++++++++++++++- latex/sections/abstract.tex | 6 ++---- latex/sections/conclusion.tex | 9 +++------ latex/sections/methods.tex | 2 +- latex/sections/results.tex | 18 ++++++++++-------- 7 files changed, 36 insertions(+), 25 deletions(-) diff --git a/latex/appendix.tex b/latex/appendix.tex index dabf25e..315fbec 100644 --- a/latex/appendix.tex +++ b/latex/appendix.tex @@ -249,14 +249,14 @@ For a particle $i$, at time step $j$, the 4th order Runge-Kutta method for a cou \end{align*} where \begin{align*} - \mathbf{k}_{\mathbf{r},1,i} &= \mathbf{v}_{i,j} \\ \mathbf{k}_{\mathbf{v},1,i} &= \frac{\mathbf{F}_{i}(t_{j}, \mathbf{v}_{i,j}, \mathbf{r}_{i,j})}{m_{i}}, \\ - \mathbf{k}_{\mathbf{r},2,i} &= \mathbf{v}_{i,j} + h \frac{\mathbf{k}_{\mathbf{v},1,i}}{2}, \\ + \mathbf{k}_{\mathbf{r},1,i} &= \mathbf{v}_{i,j}, \\ \mathbf{k}_{\mathbf{v},2,i} &= \frac{\mathbf{F}_{i}(t_{j}+\frac{h}{2}, \mathbf{v}_{i,j} + h \frac{\mathbf{k}_{\mathbf{v},1,i}}{2}, \mathbf{r}_{i,j} + h \frac{\mathbf{k}_{\mathbf{r},1,i}}{2})}{m_{i}}, \\ - \mathbf{k}_{\mathbf{r},3,i} &= \mathbf{v}_{i,j} + h \frac{\mathbf{k}_{\mathbf{v},2,i}}{2}, \\ + \mathbf{k}_{\mathbf{r},2,i} &= \mathbf{v}_{i,j} + h \frac{\mathbf{k}_{\mathbf{v},1,i}}{2}, \\ \mathbf{k}_{\mathbf{v},3,i} &= \frac{\mathbf{F}_{i}(t_{j}+\frac{h}{2}, \mathbf{v}_{i,j} + h \frac{\mathbf{k}_{\mathbf{v},2,i}}{2}, \mathbf{r}_{i,j} + h \frac{\mathbf{k}_{\mathbf{r},2,i}}{2})}{m_{i}}, \\ - \mathbf{k}_{\mathbf{r},4,i} &= \mathbf{v}_{i,j} + h \frac{\mathbf{k}_{\mathbf{v},1,i}}{2}, \\ - \mathbf{k}_{\mathbf{v},4,i} &= \frac{\mathbf{F}_{i}(t_{j}+h, \mathbf{v}_{i,j} + h \mathbf{k}_{\mathbf{v},3,i}, \mathbf{r}_{i,j} + h \mathbf{k}_{\mathbf{r},3,i})}{m_{i}}. + \mathbf{k}_{\mathbf{r},3,i} &= \mathbf{v}_{i,j} + h \frac{\mathbf{k}_{\mathbf{v},2,i}}{2}, \\ + \mathbf{k}_{\mathbf{v},4,i} &= \frac{\mathbf{F}_{i}(t_{j}+h, \mathbf{v}_{i,j} + h \mathbf{k}_{\mathbf{v},3,i}, \mathbf{r}_{i,j} + h \mathbf{k}_{\mathbf{r},3,i})}{m_{i}} \\ + \mathbf{k}_{\mathbf{r},4,i} &= \mathbf{v}_{i,j} + h \frac{\mathbf{k}_{\mathbf{v},1,i}}{2}. \end{align*} In order to find each $\mathbf{k}_{\mathbf{r},i}$ and $\mathbf{k}_{\mathbf{v},i}$, we need to first compute all $\mathbf{k}_{\mathbf{r},i}$ and $\mathbf{k}_{\mathbf{v},i}$ for all particles, then update the particle values in order to compute $\mathbf{k}_{\mathbf{r},i+1}$ and $\mathbf{k}_{\mathbf{v},i+1}$. diff --git a/latex/main.pdf b/latex/main.pdf index f61d58b642dcf2d64bee96cd13140f1b5202ef92..17fdacc9ec5c11847acee6b1e1c40a3fec108676 100644 GIT binary patch delta 189369 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z2w;hlt&~Tl)-K5ei5+6dM-lkUoj2P)y J!_Em0{TFZ^ecS*5 diff --git a/latex/references/references.bib b/latex/references/references.bib index d106fc7..666c0aa 100644 --- a/latex/references/references.bib +++ b/latex/references/references.bib @@ -101,7 +101,7 @@ author = {Ludwig-Maximilians-Universität München}, title = {Penning traps}, url = {https://www.med.physik.uni-muenchen.de/research/nuclear-science/nuclear-masses/mlltrap/layout/traps/index.html}, - urldate = {2023-10-23}, + urldate = {2023-10-23} note = {Configuration figure a} } @@ -111,3 +111,17 @@ urldate = {2023-10-24}, note = {Complete repo of project 3} } + +@online{scalasca, + author = {M. Geimer and F. Wolf and B.J.N. Wylie and E. Abraham and D. Becker and B. Mohr}, + url = {https://www.scalasca.org/scalasca/about/about.html}, + urldate = {2023-10-24}, + note = {Tool to support performance optimization of parallel programs, measuring and analyzing runtime behavior.} +} + +@online{scorep, + title = {Score-P: the Scalable Performance Measurement Infrastructure for Parallel Codes}, + url = {https://perftools.pages.jsc.fz-juelich.de/cicd/scorep/tags/latest/html/}, + urldate = {2023-10-24}, + note = {Tool suite for profiling and event tracing.} +} \ No newline at end of file diff --git a/latex/sections/abstract.tex b/latex/sections/abstract.tex index 3632e5e..cbd63f7 100644 --- a/latex/sections/abstract.tex +++ b/latex/sections/abstract.tex @@ -4,9 +4,7 @@ \begin{document} \begin{abstract} - We have studied the motion of singly-charged Calcium ions ($\text{Ca}^{+}$), inside an ideal Penning trap. With a numerical approach, studied the equations of motion by implementing the forward Euler method \(FE\) and the 4th order Runge-Kutta \(RK4\). We found that RK4 approximates the solution with smaller relative error than FE. In addition, we evaluated methods by rate of convergence. We found that RK4 has a higher convergence rate at approx. $4.0$, compared to FE at approx. $1.4$. Finally, we observed that for a time-dependent field, that there is an angular frequency that resonates with the particles in such a way that they escape the Penning trap. - - Freq. + We have studied the behavior of singly-charged Calcium ions ($\text{Ca}^{+}$), inside an ideal Penning trap. With a numerical approach, we studied the equations of motion by implementing the forward Euler method \(FE\) and the 4th order Runge-Kutta \(RK4\). We found that RK4 approximates the solution with smaller relative error than the relative error of FE. In addition, we evaluated the methods by their rate of convergence. We found that RK4 has a higher convergence rate at approx. $4.0$, compared to FE at approx. $1.4$. For particles interacting we explored angular frequencies, and amplitudes, of the time-dependent potential applied to the particles. We found that angular frequency in the range $\omega_{V} \in (1.3, 1.4)$ MHz is effective in pushing out particles, even for amplitude $f = 0.1$. \end{abstract} -\end{document} +\end{document} \ No newline at end of file diff --git a/latex/sections/conclusion.tex b/latex/sections/conclusion.tex index d0f5a1d..9469d4c 100644 --- a/latex/sections/conclusion.tex +++ b/latex/sections/conclusion.tex @@ -5,10 +5,7 @@ \section{Conclusion} We studied the movement of particles confined by an ideal Penning trap, where we used iterative methods to simulate the particle behavior. We included the magnetic and electric field of the Penning trap, in addition to simulating the particles behavior when interaction with other each other. When we introduced the interaction, the movement in both radial direction and z-direction changed. From a circular path, to a more elliptical path, where the particles initial condition determine how it is affecting other particles path. -We also compared iterative methods with the analytical solution, and found that the forward Euler \(FE\) method result in an approximation with larger relative error than the 4th order Runge-kutta \(RK4\) method. In addition to a small relative error, we also found that RK4 has a higher convergence rate at approx. $4.0$, compared to FE at approx. $1.4$. Which suggest RK4 ... +We also compared iterative methods, with the analytical solution, and found that the forward Euler \(FE\) method result in an approximation with a large relative error compared to the relative error of the 4th order Runge-kutta \(RK4\) method. In addition, we also found that RK4 has a higher convergence rate at approx. $4.0$, compared to FE at approx. $1.4$. Which suggest RK4 reach the solution faster than what FE, however, when we increase the number of time steps both methods result in similar relative error. When the number of calculations increase, and the number of time steps is sufficient, FE can be the better choise to conserve computational resources. -We finally looked at how the amount of particles that stay in the Penning trap -change when using different angular frequencies in a time-dependent field. We -observed that there seem to be a resonating frequency at around $1.4MHz$ that -makes the particles escape the Penning trap even at low amplitudes. -\end{document} +When we explored the particles behavior at angular frequencies $\omega_{V} \in (0.2, 2.5)$ MHz, we found that particles are pushed out of the Penning trap when the amplitude of the applied time-dependent potential increase. The amplitude $f = 0.7$ result in particles being pushed out at most of the range of angular frequencies, whereas an amplitude $f = 0.1$ result in particles being pushed in a more narrow range. Since particles are being pushed out when the amplitude is low, there is likely a resonance frequency at around $1.4$ MHz. +\end{document} \ No newline at end of file diff --git a/latex/sections/methods.tex b/latex/sections/methods.tex index 2cb22ae..1027824 100644 --- a/latex/sections/methods.tex +++ b/latex/sections/methods.tex @@ -253,5 +253,5 @@ The maximum error of a simulation $k$ with a step size $h_{k}$ is given by where $\mathbf{r}_{i, \text{exact}}$ is the analytical solution, and $\mathbf{r}_{i}$ is the computed result. \subsection{Tools} -The numerical methods are implemented in C++, and parallelized using \verb|OpenMP| \cite{openmp:2018}. In addition, we used profiler \verb|Scalasca| and \verb|score-p| instrumentation. We used the Python library \verb|matplotlib| \cite{hunter:2007:matplotlib} to produce all the plots, and \verb|seaborn| \cite{waskom:2021:seaborn} to set the theme in the figures. +The numerical methods are implemented in C++, and parallelized using \verb|OpenMP| \cite{openmp:2018}. In addition, we used the profiler \verb|Scalasca| \cite{scalasca} and \verb|Score-P| \cite{scorep} event tracking, during development. We used the Python library \verb|matplotlib| \cite{hunter:2007:matplotlib} to produce all the plots, and \verb|seaborn| \cite{waskom:2021:seaborn} to set the theme in the figures. \end{document} diff --git a/latex/sections/results.tex b/latex/sections/results.tex index 9f5da99..64fd5c9 100644 --- a/latex/sections/results.tex +++ b/latex/sections/results.tex @@ -96,26 +96,28 @@ In figure \ref{fig:two_particles_radial_interaction} we see the movement in radi \label{fig:3d_particles} \end{figure} -Finally, by subjecting the system to a time-dependent field, making the replacement in \ref{eq:pertubation}, we study the fraction of particles left at different amplitudes $f$. We can see how the different amplitudes lead to loss of particles, at different angular frequencies $\omega_{V}$ in \ref{fig:wide_sweep}. % Something +Finally, by subjecting the system to a time-dependent field, making the replacement in \ref{eq:pertubation}, we study the fraction of particles left at different amplitudes $f$. We can see how the different amplitudes lead to loss of particles, at different angular frequencies $\omega_{V}$ in \ref{fig:wide_sweep}. We study frequencies in the range $\omega_{V} \in (0.2, 2.5)$ MHz, with steps of $0.02$ MHz, and find that angular frequencies in the range $(1.0, 2.5)$ is effective in pushing the particles out of the Penning trap. + +We explore the range $\omega_{V} \in (1.0, 1.7)$ MHz closer in figure \ref{fig:narrow_sweep}, and observe a gradual loss of particles for amplitude $f_{1} = 0.1$. Since they are additive, a greater amplitude will result in a larger bound for the particle movement, and particles are easily pushed out. Certain angular frequencies are more effective in pushing particles out of the Penning trap, as we see in figure \ref{fig:narrower_sweep} where $\omega_{V} \in (1.3, 1.4)$ is also effective for pushing out particles of amplitude $f_{1} = 0.1$. % Something -When we study the angular frequency $\omega_{V} \in [1.1, 1.7]$ closer, in figure \ref{fig:narrow_sweep}, we observe that there are a few spots where more particles will escape. The most prominent one is where $\omega_V \in [1.1, 1.7]$, and when looking closer to the range, it seems like there's a resonating frequency at around $1.4MHz$ where All the particles will escape no matter the amplitude. When looking at the different angular frequencies with particle interaction like in \ref{fig:narrow_sweep_interactions}, we see that the amount of particles left is roughly the same as when there are no particle interactions, but that it's less predictable. \begin{figure}[H] \centering \includegraphics[width=\linewidth]{images/particles_left_wide_sweep.pdf} - \caption{Exploring particles} + \caption{Exploring the behavior of particles, where the amplitude of time-dependent potential $f = [0.1, 0.4, 0.7]$, for angular frequency $\omega_{V} \in (0.2, 2.5)$ MHz.} \label{fig:wide_sweep} \end{figure} \begin{figure}[H] \centering - \includegraphics[width=\linewidth]{images/particles_left_narrow_sweep.pdf} - \caption{Exploring different angular frequencies more closely} + \includegraphics[width=\linewidth]{images/particles_left_wide_sweep.pdf} + \caption{Exploring the behavior of particles, where the amplitude of time-dependent potential $f = [0.1, 0.4, 0.7]$, for angular frequency $\omega_{V} \in (1.0, 1.7)$ MHz.} \label{fig:narrow_sweep} \end{figure} + \begin{figure}[H] \centering - \includegraphics[width=\linewidth]{images/particles_left_narrow_sweep_interactions.pdf} - \caption{Exploring particles} - \label{fig:wide_sweep_interactions} + \includegraphics[width=\linewidth]{images/particles_left_wide_sweep.pdf} + \caption{Exploring the behavior of particles, where the amplitude of time-dependent potential $f = [0.1, 0.4, 0.7]$, for angular frequency $\omega_{V} \in (1.3, 1.4)$ MHz.} + \label{fig:narrower_sweep} \end{figure} \end{document} -- 2.20.1 From d887601c7c94707b2fdba73083d9b4af84ee5ad1 Mon Sep 17 00:00:00 2001 From: Janita Willumsen Date: Tue, 24 Oct 2023 21:49:08 +0200 Subject: [PATCH 13/23] Fix note on references --- latex/references/references.bib | 4 ++-- 1 file changed, 2 insertions(+), 2 deletions(-) diff --git a/latex/references/references.bib b/latex/references/references.bib index 666c0aa..7489417 100644 --- a/latex/references/references.bib +++ b/latex/references/references.bib @@ -102,14 +102,14 @@ title = {Penning traps}, url = {https://www.med.physik.uni-muenchen.de/research/nuclear-science/nuclear-masses/mlltrap/layout/traps/index.html}, urldate = {2023-10-23} - note = {Configuration figure a} + note = {Configuration of a Penning trap, figure a} } @online{github:repo, author = {Cory Alexander Balaton and Janita Ovidie Sandtrøen Willumsen}, url = {https://github.uio.no/FYS3150-G2-2023/Project-3}, urldate = {2023-10-24}, - note = {Complete repo of project 3} + note = {FYS3150 Project 3 repo} } @online{scalasca, -- 2.20.1 From ecf9aae9161f03ffe6e50783674f069fdf403423 Mon Sep 17 00:00:00 2001 From: Janita Willumsen Date: Tue, 24 Oct 2023 21:49:38 +0200 Subject: [PATCH 14/23] Update pdf --- latex/main.pdf | Bin 737452 -> 737452 bytes 1 file changed, 0 insertions(+), 0 deletions(-) diff --git a/latex/main.pdf b/latex/main.pdf index 17fdacc9ec5c11847acee6b1e1c40a3fec108676..b19577b0ba1f700a8bcba5faaeda441546b82aca 100644 GIT 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z?WG<@k(F#rL<(*$?0oNS8%b+Y(l+zjVp7|7H7jwOqM#KV#)AhDU`C&sD(`bo!oJQg zuk^pJEcfxhE}7}~uSn&7Y2#BJ)hadd2f}NHX_#KVbWKPm=(#ioA?_u$`zJ4Q^OHFc z@l&D}Ij{Y)zS|X6v9bh(iWiW3qcM*C#&neS*1Yci9kSAQHq7wnoLXJ)_c$IEeXrfPx{4{5@ICAEcq$puGsY=GvBX zPO@Fye3e1W3lsQ{8o84e9S<^-UYoC?F)okS@fG2G)tyr1@fm?uVJW9~)EK6H0Sni3 zuvGfwH#;b6vIG`$Ew1u@Our@(_TLm=_FgdfbMXdeW&C? za;-J5jdR?Lt5tVd)g!| Date: Tue, 24 Oct 2023 22:08:08 +0200 Subject: [PATCH 16/23] Changed some stuff --- latex/sections/results.tex | 3 +-- 1 file changed, 1 insertion(+), 2 deletions(-) diff --git a/latex/sections/results.tex b/latex/sections/results.tex index 49fb14a..1e9357e 100644 --- a/latex/sections/results.tex +++ b/latex/sections/results.tex @@ -98,8 +98,7 @@ In figure \ref{fig:two_particles_radial_interaction} we see the movement in radi Finally, by subjecting the system to a time-dependent field, making the replacement in \ref{eq:pertubation}, we study the fraction of particles left at different amplitudes $f$. We can see how the different amplitudes lead to loss of particles, at different angular frequencies $\omega_{V}$ in \ref{fig:wide_sweep}. We study frequencies in the range $\omega_{V} \in (0.2, 2.5)$ MHz, with steps of $0.02$ MHz, and find that angular frequencies in the range $(1.0, 1.7)$ is effective in pushing the particles out of the Penning trap. -We explore the range $\omega_{V} \in (1.0, 1.7)$ MHz closer in figure \ref{fig:narrow_sweep}, and observe a gradual loss of particles for amplitude $f_{1} = 0.1$. Since they are additive, a greater amplitude will result in a larger bound for the particle movement, and particles are easily pushed out. Certain angular frequencies are more effective in pushing particles out of the Penning trap, as we see in figure \ref{fig:narrow_sweep_interactions} where $\omega_{V} \in (1.3, 1.4)$ is also effective for pushing out particles of amplitude $f_{1} = 0.1$. % Something - +We explore the range $\omega_{V} \in (1.0, 1.7)$ MHz closer in figure \ref{fig:narrow_sweep}, and observe a gradual loss of particles for amplitude $f_{1} = 0.1$. Since they are additive, a greater amplitude will result in a larger bound for the particle movement, and particles are easily pushed out. Certain angular frequencies are more effective in pushing particles out of the Penning trap, when we add particle interaction. As we see in figure \ref{fig:narrow_sweep_interactions} where $\omega_{V} \in (1.3, 1.4)$ is also effective for pushing out particles of amplitude $f_{1} = 0.1$. The particle's behavior, when interactions are added, is disrupted and add to the force which result in the particle being pushed out of the Penning trap. \begin{figure}[H] \centering \includegraphics[width=\linewidth]{images/particles_left_wide_sweep.pdf} -- 2.20.1 From 30f2cda100f4a7e3c06517974893ab37f169984a Mon Sep 17 00:00:00 2001 From: Janita Willumsen Date: Tue, 24 Oct 2023 22:09:19 +0200 Subject: [PATCH 17/23] Update pdf again --- latex/main.pdf | Bin 753399 -> 746484 bytes 1 file changed, 0 insertions(+), 0 deletions(-) diff --git a/latex/main.pdf b/latex/main.pdf index 7e9dfb4cdbf98f1f03269f66e75d062a64b6dcf4..5462cf3440068f78a020c1d2a2587cd8dfcf2e75 100644 GIT binary patch delta 138291 zcmV)4K+3=O{VnvzEf6J8L`E$!E;R}z(A34)3Ne=v2@4YhG&M6hlhH~jf3;g%bK6Fi ze)q57zU)A?VHwlY;VlwUM$$KCBcmpur??qRvY(QVaj>NWR&SM65K zdC!feJ@j0|D}7fnr>t9Ks`Q@dY<#9P5ZZ$9mVDsgu_<<`yvQAqduP%~`iv!;Bysx`vMS6iF z$79!4D_nYuyf7~LAY{d1ChHm-CDRmp>)+3yh(r`@A}w)z-<}%a04f7E-{CqBTyOTr 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We can see how the different amplitudes lead to loss of particles, at different angular frequencies $\omega_{V}$ in \ref{fig:wide_sweep}. We study frequencies in the range $\omega_{V} \in (0.2, 2.5)$ MHz, with steps of $0.02$ MHz, and find that angular frequencies in the range $(1.0, 1.7)$ is effective in pushing the particles out of the Penning trap. -We explore the range $\omega_{V} \in (1.0, 1.7)$ MHz closer in figure \ref{fig:narrow_sweep}, and observe a gradual loss of particles for amplitude $f_{1} = 0.1$. Since they are additive, a greater amplitude will result in a larger bound for the particle movement, and particles are easily pushed out. Certain angular frequencies are more effective in pushing particles out of the Penning trap, when we add particle interaction. As we see in figure \ref{fig:narrow_sweep_interactions} where $\omega_{V} \in (1.3, 1.4)$ is also effective for pushing out particles of amplitude $f_{1} = 0.1$. When particle interactions are added, we see that there tends to be less particles left in the Penning trap than without any interactions. We also see that the amount of particles left is more unpredictable when there are particle interactions. +We explore the range $\omega_{V} \in (1.0, 1.7)$ MHz closer in figure \ref{fig:narrow_sweep}, and observe a gradual loss of particles for amplitude $f_{1} = 0.1$. Since they are additive, a greater amplitude will result in a larger bound for the particle movement, and particles are easily pushed out. Certain angular frequencies are more effective in pushing particles out of the Penning trap, when we add particle interaction. As we see in figure \ref{fig:narrow_sweep_interactions} where $\omega_{V} \in (1.3, 1.4)$ is also effective for pushing out particles of amplitude $f_{1} = 0.1$. When particle interactions are added, we see that there tends to be fewer particles left in the Penning trap than without any interactions. We also see that the amount of particles left is more unpredictable when there are particle interactions. %The particles' behavior, when interactions are added, is disrupted and add to the force which result in the particle being pushed out of the Penning trap. \begin{figure}[H] \centering -- 2.20.1 From 82cdbaced53ab2d6bcd7f5f24262a3f1b3261478 Mon Sep 17 00:00:00 2001 From: Janita Willumsen Date: Tue, 24 Oct 2023 22:32:53 +0200 Subject: [PATCH 20/23] Final, hopefully, changes --- latex/main.pdf | Bin 753602 -> 746541 bytes 1 file changed, 0 insertions(+), 0 deletions(-) diff --git a/latex/main.pdf b/latex/main.pdf index 5c4a2a88bf80a9158988e6f3537ef201c4501bf5..03e7eef8a9bb2336bdd5fa1b010830b4098e1123 100644 GIT binary patch delta 138392 zcmV)GK)%1i|1GV^Ehi;VL`E$!E;R}z(A34)3NbJsFd%PYY6?6&3NK7$ZfA68AT%{I zIg`;!D1WtETXWk+mVWoI;J)lYwP7250UAgun~Gy8QBoOuZ0$^KCD|&Nghrqzi@WvWm*j1;zp^?x@2XRC*jz2G&yv?y)_){bcWPF@Y7TGYieCBzxhYA+nx74H0dN+Iyni@ 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We can see how the different amplitudes lead to loss of particles, at different angular frequencies $\omega_{V}$ in \ref{fig:wide_sweep}. We study frequencies in the range $\omega_{V} \in (0.2, 2.5)$ MHz, with steps of $0.02$ MHz, and find that angular frequencies in the range $(1.0, 1.7)$ is effective in pushing the particles out of the Penning trap. +Finally, by subjecting the system to a time-dependent field, making the replacement in \ref{eq:pertubation}, we study the fraction of particles left at different amplitudes $f$. We can see how the different amplitudes lead to loss of particles, at different angular frequencies $\omega_{V}$ in \ref{fig:wide_sweep}. We study frequencies in the range $\omega_{V} \in (0.2, 2.5)$ MHz, with steps of $0.02$ MHz, and find that angular frequencies in the range $(1.0, 2.5)$ is effective in pushing the particles out of the Penning trap. + +We explore the range $\omega_{V} \in (1.0, 1.7)$ MHz closer in figure \ref{fig:narrow_sweep}, and observe a loss of particles for amplitude $f_{1} = 0.1$ in the range $\omega_{V} \in (1.3, 1.5)$, for $f_{2} = 0.4$ in the range $\omega_{V} \in (1.2, 1.6)$, and $f_{3} = 0.7$ in the entire range explored. When we introduce particle interaction, we observe the amplitude with higher value will result in a larger bound for the particle movement, and particles are easily pushed out. When we add particle interaction the angular frequencies, which result in particles being pushed out, are similar as in no interaction. As we see in figure \ref{fig:narrow_sweep_interactions} the particle's behavior, when interactions are added, is disrupted and add to the force resulting in the particle being pushed out of the Penning trap. We see that fewer particles tend to be left in the Penning trap, than without any particle interactions. We also see that the fraction of particles left is more unpredictable with particle interactions. -We explore the range $\omega_{V} \in (1.0, 1.7)$ MHz closer in figure \ref{fig:narrow_sweep}, and observe a gradual loss of particles for amplitude $f_{1} = 0.1$. Since they are additive, a greater amplitude will result in a larger bound for the particle movement, and particles are easily pushed out. Certain angular frequencies are more effective in pushing particles out of the Penning trap, when we add particle interaction. As we see in figure \ref{fig:narrow_sweep_interactions} where $\omega_{V} \in (1.3, 1.4)$ is also effective for pushing out particles of amplitude $f_{1} = 0.1$. When particle interactions are added, we see that there tends to be fewer particles left in the Penning trap than without any interactions. We also see that the amount of particles left is more unpredictable when there are particle interactions. -%The particles' behavior, when interactions are added, is disrupted and add to the force which result in the particle being pushed out of the Penning trap. \begin{figure}[H] \centering \includegraphics[width=\linewidth]{images/particles_left_wide_sweep.pdf} -- 2.20.1 From 09d81d59177358545a9c4ad2bc4465337db12eb6 Mon Sep 17 00:00:00 2001 From: Cory Date: Tue, 24 Oct 2023 22:42:13 +0200 Subject: [PATCH 22/23] Final edition of the report --- latex/main.pdf | Bin 746691 -> 753758 bytes 1 file changed, 0 insertions(+), 0 deletions(-) diff --git a/latex/main.pdf b/latex/main.pdf index 59829820a78c34dc8d8da63020d9749cff3dbcf6..962bca866eaa28ccd26a52f9ba385aa161735575 100644 GIT binary patch delta 145586 zcmV)XK&`*S$t~W1E)peBL`E$!E;R}z(A34)3NbM@kr5OEH8_*;ktlz)JKK`mHkR-H z3hm1dyC#B(JDF6rDvs^MNoDM{duM7Z$yTvNc8fcbsF0HF_Sf&Z0KCZPwC7=~9{?PH 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z5wxVnAnow~3L2PhUXP(#bU&Dz7^B1(A6tjaUz4n5fFAz82e9vL3$I_0kFyJz?Cr~{ Om@FALw}ycx+5Z67 Date: Tue, 24 Oct 2023 22:44:55 +0200 Subject: [PATCH 23/23] Add seaborn --- requirements.txt | 1 + 1 file changed, 1 insertion(+) diff --git a/requirements.txt b/requirements.txt index db5d81e..4a00803 100644 --- a/requirements.txt +++ b/requirements.txt @@ -1,2 +1,3 @@ matplotlib numpy +seaborn -- 2.20.1

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