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 f61d58b..17fdacc 100644
Binary files a/latex/main.pdf and b/latex/main.pdf differ
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}