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z%(id(Ps`j_G)gwE3~)&p=u+J8E4>_L(5-qO1uI_7au+CH%wCHW56@hD%J&*PZ&N6_ zyJ?dt-Z`AnxxR7dX68svE*bJV9rMB+@$yYd_WAz!NOOH`w09pfn%O@(%3m;@^2im} zdUc;{s6p-cOOQT#4ILleg>T_i|6Ztd{VZ7V^Lv@i;2XHFsNv$@TehQ Date: Fri, 20 Oct 2023 15:58:49 +0200 Subject: [PATCH 02/15] Latexify the plots --- src/scripts/plot_2_particles.py | 20 ++++++++++++++++--- src/scripts/plot_3d.py | 14 +++++++++++++ src/scripts/plot_particles_left.py | 14 +++++++++++++ src/scripts/plot_phase_space.py | 18 +++++++++++++++-- src/scripts/plot_relative_error.py | 24 +++++++++++++++++----- src/scripts/plot_single_particle.py | 31 +++++++++++++++++++++++------ 6 files changed, 105 insertions(+), 16 deletions(-) diff --git a/src/scripts/plot_2_particles.py b/src/scripts/plot_2_particles.py index 17f327c..7acfa50 100644 --- a/src/scripts/plot_2_particles.py +++ b/src/scripts/plot_2_particles.py @@ -1,5 +1,19 @@ import matplotlib.pyplot as plt import numpy as np +import seaborn as sns + +sns.set_theme() +params = { + "font.family": "Serif", + "font.serif": "Roman", + "text.usetex": True, + "axes.titlesize": "large", + "axes.labelsize": "large", + "xtick.labelsize": "large", + "ytick.labelsize": "large", + "legend.fontsize": "medium" +} +plt.rcParams.update(params) def main(): files = [ @@ -29,10 +43,10 @@ def main(): plt.xlabel(r"x $(\mu m)$") plt.ylabel(r"y $(\mu m)$") - plt.title(r"2 particles with and without interactions.") - # plt.legend() - # plt.show() + # plt.title(r"2 particles with and without interactions.") + plt.legend(loc="upper right") plt.savefig("../latex/images/plot_2_particles_xy.pdf") + plt.show() if __name__ == "__main__": diff --git a/src/scripts/plot_3d.py b/src/scripts/plot_3d.py index 811bf6a..2dbbdd4 100644 --- a/src/scripts/plot_3d.py +++ b/src/scripts/plot_3d.py @@ -1,5 +1,19 @@ import matplotlib.pyplot as plt import numpy as np +import seaborn as sns + +sns.set_theme() +params = { + "font.family": "Serif", + "font.serif": "Roman", + "text.usetex": True, + "axes.titlesize": "large", + "axes.labelsize": "large", + "xtick.labelsize": "large", + "ytick.labelsize": "large", + "legend.fontsize": "medium" +} +plt.rcParams.update(params) def main(): files = [ diff --git a/src/scripts/plot_particles_left.py b/src/scripts/plot_particles_left.py index 85bfd78..42e65b9 100644 --- a/src/scripts/plot_particles_left.py +++ b/src/scripts/plot_particles_left.py @@ -1,4 +1,18 @@ import matplotlib.pyplot as plt +import seaborn as sns + +sns.set_theme() +params = { + "font.family": "Serif", + "font.serif": "Roman", + "text.usetex": True, + "axes.titlesize": "large", + "axes.labelsize": "large", + "xtick.labelsize": "large", + "ytick.labelsize": "large", + "legend.fontsize": "medium" +} +plt.rcParams.update(params) def main(): with open("output/time_dependent_potential/res.txt") as f: diff --git a/src/scripts/plot_phase_space.py b/src/scripts/plot_phase_space.py index 5a4c3ac..849dbfe 100644 --- a/src/scripts/plot_phase_space.py +++ b/src/scripts/plot_phase_space.py @@ -1,5 +1,19 @@ import matplotlib.pyplot as plt import numpy as np +import seaborn as sns + +sns.set_theme() +params = { + "font.family": "Serif", + "font.serif": "Roman", + "text.usetex": True, + "axes.titlesize": "large", + "axes.labelsize": "large", + "xtick.labelsize": "large", + "ytick.labelsize": "large", + "legend.fontsize": "medium" +} +plt.rcParams.update(params) def main(): directories = { @@ -21,7 +35,7 @@ def main(): r"particle 1 v", r"particle 2 r", r"particle 2 v", - ] + ] colors = [ "lightskyblue", "deepskyblue", @@ -44,9 +58,9 @@ def main(): axs1[i].legend() axs1[i].set_title(title) - # plt.show() fig1.savefig("../latex/images/phase_space_2_particles_x.pdf") fig2.savefig("../latex/images/phase_space_2_particles_z.pdf") + plt.show() if __name__ == "__main__": diff --git a/src/scripts/plot_relative_error.py b/src/scripts/plot_relative_error.py index 54214c8..10106af 100644 --- a/src/scripts/plot_relative_error.py +++ b/src/scripts/plot_relative_error.py @@ -1,5 +1,19 @@ import matplotlib.pyplot as plt import numpy as np +import seaborn as sns + +sns.set_theme() +params = { + "font.family": "Serif", + "font.serif": "Roman", + "text.usetex": True, + "axes.titlesize": "large", + "axes.labelsize": "large", + "xtick.labelsize": "large", + "ytick.labelsize": "large", + "legend.fontsize": "medium" +} +plt.rcParams.update(params) def main(): directories = { @@ -22,7 +36,7 @@ def main(): "Relative error for the RK4 method", "Relative error for the forward Euler method" ] - fig1, axs1 = plt.subplots(2,1) + fig1, axs1 = plt.subplots(2,1, sharex=True) for i, (dir, title) in enumerate(zip(directories, titles)): max_err = [] for label, file in zip(labels, files): @@ -33,17 +47,17 @@ def main(): max_err.append(max(r)) axs1[i].plot(t, r, label=label) - axs1[i].set(xlabel=r"t $(\mu s)$", ylabel = r"relative_error $(\mu m)$") + axs1[i].set(ylabel = r"relative_error $(\mu m)$") # xlabel=r"t $(\mu s)$", - + axs1[1].set_xlabel(r"t $(\mu s)$") axs1[i].legend() - axs1[i].set_title(title) + # axs1[i].set_title(title) conv_rate = 1/3 * sum([np.log10(max_err[i+1]/max_err[i])/np.log10(.5) for i in range(3)]) print(conv_rate) - plt.show() # fig1.savefig("../latex/images/phase_space_2_particles_x.pdf") + plt.show() if __name__ == "__main__": diff --git a/src/scripts/plot_single_particle.py b/src/scripts/plot_single_particle.py index f4bdc6a..77ea776 100644 --- a/src/scripts/plot_single_particle.py +++ b/src/scripts/plot_single_particle.py @@ -1,5 +1,19 @@ import matplotlib.pyplot as plt import numpy as np +import seaborn as sns + +sns.set_theme() +params = { + "font.family": "Serif", + "font.serif": "Roman", + "text.usetex": True, + "axes.titlesize": "large", + "axes.labelsize": "large", + "xtick.labelsize": "large", + "ytick.labelsize": "large", + "legend.fontsize": "medium" +} +plt.rcParams.update(params) def z(t): V_0 = 25.*9.64852558 * 10**4 @@ -9,20 +23,25 @@ def z(t): return 20.*np.cos(w_z*t) def main(): + colors = [ + "lightskyblue", + "deepskyblue", + "salmon", + "tomato", + ] filename = "output/simulate_single_particle/particle_0_r.txt" r = t = [] with open(filename) as f: lines = f.readlines() t = np.linspace(0, 50, len(lines)) r = np.array([list(map(float, line.strip().split(","))) for line in lines]) - - plt.plot(t, r[:, 2], label="approximation") - plt.plot(t, z(t), label="analytical") + plt.plot(t, r[:, 2], label="approximation", color=colors[1]) + plt.plot(t, z(t), label="analytical", color=colors[2], linestyle="dotted") plt.xlabel(r"time $(\mu s)$") plt.ylabel(r"z $(\mu m)$") - plt.title(r"Movement of a single particle in the x direction") - plt.legend() - # plt.savefig("../latex/images/single_particle.pdf") + # plt.title(r"Movement of a single particle in the x direction") + plt.legend(loc="lower right") + plt.savefig("../latex/images/single_particle.pdf") plt.show() From 405493a88891d138f49bfa5387dc9e059ad0ae69 Mon Sep 17 00:00:00 2001 From: Janita Willumsen Date: Fri, 20 Oct 2023 15:59:39 +0200 Subject: 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Subject: [PATCH 04/15] Update plot with seaborn theme --- src/scripts/plot_2_particles.py | 2 +- src/scripts/plot_particles_left.py | 31 ++++++++++++++++++++--------- src/scripts/plot_single_particle.py | 20 ++++++++++++------- 3 files changed, 36 insertions(+), 17 deletions(-) diff --git a/src/scripts/plot_2_particles.py b/src/scripts/plot_2_particles.py index 7acfa50..be7838b 100644 --- a/src/scripts/plot_2_particles.py +++ b/src/scripts/plot_2_particles.py @@ -45,7 +45,7 @@ def main(): plt.ylabel(r"y $(\mu m)$") # plt.title(r"2 particles with and without interactions.") plt.legend(loc="upper right") - plt.savefig("../latex/images/plot_2_particles_xy.pdf") + plt.savefig("../latex/images/plot_2_particles_xy.pdf", bbox_inches="tight") plt.show() diff --git a/src/scripts/plot_particles_left.py b/src/scripts/plot_particles_left.py index 42e65b9..9884cd2 100644 --- a/src/scripts/plot_particles_left.py +++ b/src/scripts/plot_particles_left.py @@ -15,6 +15,14 @@ params = { plt.rcParams.update(params) def main(): + colors = [ + "lightskyblue", + "deepskyblue", + "salmon", + "tomato", + "mediumaquamarine", + "mediumseagreen" + ] with open("output/time_dependent_potential/res.txt") as f: lines = f.readlines() x = [] @@ -28,17 +36,22 @@ def main(): y2.append(float(l[2])) y3.append(float(l[3])) - plt.plot(x,y1,label=f"amplitude: 0.1") - plt.plot(x,y2,label=f"amplitude: 0.4") - plt.plot(x,y3,label=f"amplitude: 0.7") + fig, ax = plt.subplots() + ax.plot(x, y1, label=r"$f_{1} = 0.1$", color=colors[0]) + ax.plot(x, y2, label=r"$f_{2} = 0.4$", color=colors[2]) + ax.plot(x, y3, label=r"$f_{3} = 0.7$", color=colors[4]) + + ax.set_xlabel(r"Frequency $\omega_V$ (MHz)") + ax.set_xlim((0, 2.8)) + ax.set_ylabel(r"Fraction of particles left") + ax.set_ylim((-0.1, 1.1)) + # plt.title(r"The fraction of particles left in the Penning trap " + # "after 500 microseconds for different amplitudes and frequencies") + + ax.legend(loc="upper right") - plt.xlabel(r"$\omega_V$ (MHz)") - plt.ylabel(r"Fraction of particles left") - plt.title(r"The fraction of particles left in the Penning trap " - "after 500 microseconds for different amplitudes and frequencies") - plt.legend() # plt.show() - plt.savefig("../latex/images/particles_left.pdf") + fig.savefig("../latex/images/particles_left.pdf", bbox_inches="tight") if __name__ == "__main__": main() diff --git a/src/scripts/plot_single_particle.py b/src/scripts/plot_single_particle.py index 77ea776..6ee2d16 100644 --- a/src/scripts/plot_single_particle.py +++ b/src/scripts/plot_single_particle.py @@ -28,6 +28,8 @@ def main(): "deepskyblue", "salmon", "tomato", + "mediumaquamarine", + "mediumseagreen" ] filename = "output/simulate_single_particle/particle_0_r.txt" r = t = [] @@ -35,14 +37,18 @@ def main(): lines = f.readlines() t = np.linspace(0, 50, len(lines)) r = np.array([list(map(float, line.strip().split(","))) for line in lines]) - plt.plot(t, r[:, 2], label="approximation", color=colors[1]) - plt.plot(t, z(t), label="analytical", color=colors[2], linestyle="dotted") - plt.xlabel(r"time $(\mu s)$") - plt.ylabel(r"z $(\mu m)$") + + fig, ax = plt.subplots() + ax.plot(t, r[:, 2], label="approximation", color=colors[3]) + ax.plot(t, z(t), label="analytical", color=colors[5], linestyle="dotted") + ax.set_xlabel(r"Time $(\mu s)$") + ax.set_xlim((-5, 55)) + ax.set_ylabel(r"z $(\mu m)$") + ax.set_ylim((-25, 25)) # plt.title(r"Movement of a single particle in the x direction") - plt.legend(loc="lower right") - plt.savefig("../latex/images/single_particle.pdf") - plt.show() + ax.legend(loc="upper right") + fig.savefig("../latex/images/single_particle.pdf", bbox_inches="tight") + # plt.show() if __name__ == "__main__": From 1918dfb3f7da48096ad84472a2e175d8990b384d Mon Sep 17 00:00:00 2001 From: Janita Willumsen Date: Sun, 22 Oct 2023 14:22:44 +0200 Subject: [PATCH 05/15] Update plots --- latex/images/particles_left.pdf | Bin 17840 -> 135426 bytes latex/images/single_particle.pdf | Bin 104957 -> 104683 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zhniXcf0jct_DO7YbERl!24$;5EJ;VBXE&d3$jz~K+RcLHC~%CuR~Bu*tKxG*f?of8 z@w9Ls*$;gDzSn;GK5~vb0Ka!LaV41KlQ`s;%)YL8_kR9U*ZU{azRy3#G$BK6k!}}1 zv$Ymr`!4mSXpf@?X1$z(Q2{YznvDth7l}O=&(>8LOQ6{*kv#+C(|2U^bxt{*EsSsp zr3-Gq;QJWKm!k*T?OOp1Gi7+aNoan`@8>CZiRUS3-=EFL^i|C*u^xLb|K~iG7|n}i z)z#YK(kUnI+v_(kGTbTFNAOyWkgM)8h?LNAw)3|-HsheA)8_5q`@26c@Tob;=cmUL z8_hpx5m}PF8I)9%$U-0?aD)g|n+O$x#CtRlj{rYNfDa1EMo@+D^YHz%2>iF`+~_d> zJ$XQUU@+)Em@psz-;5t5B>109Sm-|(4-feNvVZ3Y3jD*M%1{J6lE0-vARz$|80rQ^ zV1n?2|I+*SlJ{>teqKJIfA#*U|M39=fkA?Q`S^$N3jg(I5I^rf!~S|J#K#Bz?>^Q4 zCCJYQ;^+CRk)MxO@UH+tLVN;$8~!I$A%3W54gwLo;NKEL{Jj5PX2QbI+#Cc&Og;fY MOeQ9IO$E&V0tLMh=>Px# From eab84f52e95ebb89145e4dd25ad7a4f23b51f53e Mon Sep 17 00:00:00 2001 From: Cory Date: Sun, 22 Oct 2023 14:32:29 +0200 Subject: [PATCH 06/15] Stuff --- src/scripts/plot_3d.py | 3 ++- 1 file changed, 2 insertions(+), 1 deletion(-) diff --git a/src/scripts/plot_3d.py b/src/scripts/plot_3d.py index 811bf6a..535a27a 100644 --- a/src/scripts/plot_3d.py +++ b/src/scripts/plot_3d.py @@ -16,8 +16,8 @@ def main(): ] colors = [ "lightskyblue", - "deepskyblue", "salmon", + "deepskyblue", "darkred" ] ax = plt.figure().add_subplot(projection="3d") @@ -31,6 +31,7 @@ def main(): ax.set_xlabel(r"x $(\mu m)$") ax.set_ylabel(r"y $(\mu m)$") ax.set_zlabel(r"z $(\mu m)$") + ax.view_init(10,45) plt.title(r"2 particles with and without interactions.") plt.legend() plt.savefig("../latex/images/3d_plot.pdf") From 0dfda79fda6d2dfffb82475f7758c647c698ac01 Mon Sep 17 00:00:00 2001 From: Cory Date: Sun, 22 Oct 2023 15:09:14 +0200 Subject: [PATCH 07/15] Update plots --- src/scripts/plot_2_particles.py | 17 ++++++++--------- src/scripts/plot_3d.py | 18 ++++++++++-------- 2 files changed, 18 insertions(+), 17 deletions(-) diff --git a/src/scripts/plot_2_particles.py b/src/scripts/plot_2_particles.py index be7838b..92657ae 100644 --- a/src/scripts/plot_2_particles.py +++ b/src/scripts/plot_2_particles.py @@ -23,16 +23,16 @@ def main(): "output/simulate_2_particles/with_interaction/particle_1_r.txt" ] labels = [ - "particle 1 no interaction", - "particle 2 no interaction", - "particle 1 with interaction", - "particle 2 with interaction", + r"$p_1$", + r"$p_2$", + r"$\hat{p}_1$", + r"$\hat{p}_2$", ] colors = [ - "lightskyblue", - "lightskyblue", + "mediumaquamarine", "salmon", - "salmon" + "seagreen", + "darkred" ] for label, color, file in zip(labels, colors, files): with open(file) as f: @@ -43,10 +43,9 @@ def main(): plt.xlabel(r"x $(\mu m)$") plt.ylabel(r"y $(\mu m)$") - # plt.title(r"2 particles with and without interactions.") plt.legend(loc="upper right") + plt.axis("equal") plt.savefig("../latex/images/plot_2_particles_xy.pdf", bbox_inches="tight") - plt.show() if __name__ == "__main__": diff --git a/src/scripts/plot_3d.py b/src/scripts/plot_3d.py index 6aad06c..4134787 100644 --- a/src/scripts/plot_3d.py +++ b/src/scripts/plot_3d.py @@ -11,7 +11,8 @@ params = { "axes.labelsize": "large", "xtick.labelsize": "large", "ytick.labelsize": "large", - "legend.fontsize": "medium" + "legend.fontsize": "medium", + "figure.autolayout": True } plt.rcParams.update(params) @@ -23,15 +24,15 @@ def main(): "output/simulate_2_particles/with_interaction/particle_1_r.txt" ] labels = [ - "particle 1 no interaction", - "particle 2 no interaction", - "particle 1 with interaction", - "particle 2 with interaction", + r"$p_1$", + r"$p_2$", + r"$\hat{p}_1$", + r"$\hat{p}_2$", ] colors = [ - "lightskyblue", + "mediumaquamarine", "salmon", - "deepskyblue", + "seagreen", "darkred" ] ax = plt.figure().add_subplot(projection="3d") @@ -45,7 +46,8 @@ def main(): ax.set_xlabel(r"x $(\mu m)$") ax.set_ylabel(r"y $(\mu m)$") ax.set_zlabel(r"z $(\mu m)$") - ax.view_init(10,45) + ax.view_init(10,-35) + plt.title(r"2 particles with and without interactions.") plt.legend() plt.savefig("../latex/images/3d_plot.pdf") From e42c108d202d699a9e50e11e720cf2ce70517be9 Mon Sep 17 00:00:00 2001 From: Cory Date: Sun, 22 Oct 2023 15:19:45 +0200 Subject: [PATCH 08/15] Made some changes --- src/scripts/plot_2_particles.py | 5 +++++ 1 file changed, 5 insertions(+) diff --git a/src/scripts/plot_2_particles.py b/src/scripts/plot_2_particles.py index 92657ae..486f107 100644 --- a/src/scripts/plot_2_particles.py +++ b/src/scripts/plot_2_particles.py @@ -34,13 +34,18 @@ def main(): "seagreen", "darkred" ] + start_pos = set() for label, color, file in zip(labels, colors, files): with open(file) as f: lines = f.readlines() t = np.linspace(0, 50, len(lines)) r = np.array([list(map(float, line.strip().split(","))) for line in lines]) plt.plot(r[:,0], r[:,1], label=label, color=color) + start_pos.add((r[0,0],r[0,1])) + for pos in start_pos: + plt.plot(*pos, "o", color="black") + plt.xlabel(r"x $(\mu m)$") plt.ylabel(r"y $(\mu m)$") plt.legend(loc="upper right") From 8373deea81125611251d855b732359d28c358b39 Mon Sep 17 00:00:00 2001 From: Cory Date: Sun, 22 Oct 2023 16:34:48 +0200 Subject: [PATCH 09/15] Update plots --- src/scripts/plot_phase_space.py | 63 ++++++++++++++++------------- src/scripts/plot_single_particle.py | 6 +-- 2 files changed, 39 insertions(+), 30 deletions(-) diff --git a/src/scripts/plot_phase_space.py b/src/scripts/plot_phase_space.py index 849dbfe..b989961 100644 --- a/src/scripts/plot_phase_space.py +++ b/src/scripts/plot_phase_space.py @@ -20,21 +20,13 @@ def main(): "output/simulate_2_particles/no_interaction/", "output/simulate_2_particles/with_interaction/", } - titles = { - "particles without interaction", - "particles with interaction" - } files = [ - "particle_0_r.txt", - "particle_0_v.txt", - "particle_1_r.txt", - "particle_1_v.txt", + "particle_0", + "particle_1", ] labels = [ - r"particle 1 r", - r"particle 1 v", - r"particle 2 r", - r"particle 2 v", + [r"$p_1$", r"$\hat{p}_1$"], + [r"$p_2$", r"$\hat{p}_2$"] ] colors = [ "lightskyblue", @@ -42,25 +34,42 @@ def main(): "salmon", "tomato", ] - fig1, axs1 = plt.subplots(2,1) - fig2, axs2 = plt.subplots(2,1) - for i, (dir, title) in enumerate(zip(directories, titles)): - for label, color, file in zip(labels, colors, files): - with open(dir+file) as f: + linestyles = [ + "solid", + "dotted" + ] + fig1, axs1 = plt.subplots(2,1,sharex=True) + fig2, axs2 = plt.subplots(2,1,sharex=True) + for i, (file, label) in enumerate(zip(files, labels)): + for j, (dir, linestyle) in enumerate(zip(directories, linestyles)): + with open(dir+file+"_r.txt") as f: lines = f.readlines() t = np.linspace(0, 50, len(lines)) r = np.array([list(map(float, line.strip().split(","))) for line in lines]) - axs1[i].plot(t, r[:,0], label=label, color=color) - axs2[i].plot(t, r[:,2], label=label, color=color) - - axs1[i].set(xlabel=r"t $(\mu s)$", ylabel = r"z $(\mu m)$") - - axs1[i].legend() - axs1[i].set_title(title) + axs1[i].plot(t, r[:,0], label=f"{label[j]} r", linestyle=linestyle) + axs2[i].plot(t, r[:,2], label=f"{label[j]} r", linestyle=linestyle) - fig1.savefig("../latex/images/phase_space_2_particles_x.pdf") - fig2.savefig("../latex/images/phase_space_2_particles_z.pdf") - plt.show() + with open(dir+file+"_v.txt") as f: + lines = f.readlines() + t = np.linspace(0, 50, len(lines)) + r = np.array([list(map(float, line.strip().split(","))) for line in lines]) + axs1[i].plot(t, r[:,0], label=f"{label[j]} v", linestyle=linestyle) + axs2[i].plot(t, r[:,2], label=f"{label[j]} v", linestyle=linestyle) + + + axs1[1].set(xlabel=r"t $(\mu s)$", ylabel = r"x $(\mu m)$") + axs1[0].set(ylabel = r"x $(\mu m)$") + axs2[1].set(xlabel=r"t $(\mu s)$", ylabel = r"z $(\mu m)$") + axs2[0].set(ylabel = r"z $(\mu m)$") + + axs1[i].legend() + # axs1[i].set_title(title) + axs2[i].legend() + # axs2[i].set_title(title) + + fig1.savefig("../latex/images/phase_space_2_particles_x.pdf", bbox_inches="tight") + fig2.savefig("../latex/images/phase_space_2_particles_z.pdf", bbox_inches="tight") + # plt.show() if __name__ == "__main__": diff --git a/src/scripts/plot_single_particle.py b/src/scripts/plot_single_particle.py index 6ee2d16..a80a8b4 100644 --- a/src/scripts/plot_single_particle.py +++ b/src/scripts/plot_single_particle.py @@ -39,9 +39,9 @@ def main(): r = np.array([list(map(float, line.strip().split(","))) for line in lines]) fig, ax = plt.subplots() - ax.plot(t, r[:, 2], label="approximation", color=colors[3]) - ax.plot(t, z(t), label="analytical", color=colors[5], linestyle="dotted") - ax.set_xlabel(r"Time $(\mu s)$") + ax.plot(t, r[:, 2], label="approximation", color="mediumseagreen") + ax.plot(t, z(t), label="analytical", color="black", linestyle="dotted") + ax.set_xlabel(r"t $(\mu s)$") ax.set_xlim((-5, 55)) ax.set_ylabel(r"z $(\mu m)$") ax.set_ylim((-25, 25)) From 493bac88c34c8913da8f4190e48556b86bdbfab6 Mon Sep 17 00:00:00 2001 From: Janita Willumsen Date: Sun, 22 Oct 2023 17:14:03 +0200 Subject: [PATCH 10/15] Update document --- latex/appendix/appendix.tex | 44 ------- latex/appendix/appendix_a.tex | 29 +++++ latex/appendix/appendix_b.tex | 145 +++++++++++++++++++++ latex/appendix/appendix_c.tex | 21 +++ latex/images/3d_plot.pdf | Bin 132527 -> 159580 bytes latex/images/particles_left.pdf | Bin 135426 -> 135426 bytes latex/images/phase_space_2_particles_x.pdf | Bin 137791 -> 169476 bytes latex/images/phase_space_2_particles_z.pdf | Bin 82638 -> 151392 bytes latex/images/plot_2_particles_xy.pdf | Bin 157214 -> 184652 bytes latex/images/relative_error.pdf | Bin 0 -> 165973 bytes latex/main.pdf | Bin 0 -> 673380 bytes latex/main.tex | 13 +- latex/sections/introduction.tex | 4 +- latex/sections/methods.tex | 116 +++++++++++++---- latex/sections/results.tex | 83 +++++++++++- 15 files changed, 373 insertions(+), 82 deletions(-) delete mode 100644 latex/appendix/appendix.tex create mode 100644 latex/appendix/appendix_a.tex create mode 100644 latex/appendix/appendix_b.tex create mode 100644 latex/appendix/appendix_c.tex create mode 100644 latex/images/relative_error.pdf create mode 100644 latex/main.pdf diff --git a/latex/appendix/appendix.tex b/latex/appendix/appendix.tex deleted file mode 100644 index eac7190..0000000 --- a/latex/appendix/appendix.tex +++ /dev/null @@ -1,44 +0,0 @@ -\documentclass[../main.tex]{subfiles} -\graphicspath{{\subfix{../images/}}} - -\begin{document} -\appendix -\section*{Appendix A}\label{sec:appendix_a} -Equations given -\begin{equation}\label{eq:newton_second} - m \ddot{\mathbf{r}} = \sum_{i} \mathbf{F}_{i} -\end{equation} -% - -% -\begin{equation}\label{eq:e_field_point} - \mathbf{E} = k_{e} \sum_{j=1}^{n} q_j \frac{\mathbf{r} - \mathbf{r}_{j}}{|\mathbf{r} - \mathbf{r}_{j}|^{3}} -\end{equation} -% -\begin{equation}\label{eq:e_field_potential} - \mathbf{E} = - \nabla V -\end{equation} - -\section*{Appendix B}\label{sec:appendix_b} -Sum of all forces -\begin{align*} - sum -\end{align*} - -We find the physical coordinates from $x(t) = \text{Re} f(t)$ and $y(t) = \text{Im} f(t)$, where $f(t)$ is given by equation \eqref{eq:general_solution}. - -We can rewrite $f(t)$ using the definition as -\begin{align*} - % f(t) =& A_{+}e^{-i(\omega_{+} t + \phi_{+})} + A_{-}e^{-i(\omega_{-} t + \phi_{-})} \\ - f(t) =& A_{+}(\cos{\omega_{+} t + \phi_{+}} - i \sin{\omega_{+} t + \phi_{+}}) \\ - \numberthis \label{eq:general_solution_trig} - &+ A_{-}(\cos{\omega_{-} t + \phi_{-}} - i \sin{\omega_{-} t + \phi_{-}}) -\end{align*} -% -If we rearrange the right side of equation \eqref{eq:general_solution_trig}, we find the physical coordinates -\begin{align}\label{eq:physical_coord} - x(t) &= A_{+}(\cos{\omega_{+} t + \phi_{+}}) + A_{-}(\cos{\omega_{-} t + \phi_{-}}) \\ - y(t) &= - A_{+}(i \sin{\omega_{+} t + \phi_{+}}) - A_{-}(i \sin{\omega_{-} t + \phi_{-}}) -\end{align} - -\end{document} \ No newline at end of file diff --git a/latex/appendix/appendix_a.tex b/latex/appendix/appendix_a.tex new file mode 100644 index 0000000..ccc0a85 --- /dev/null +++ b/latex/appendix/appendix_a.tex @@ -0,0 +1,29 @@ +\documentclass[../main.tex]{subfiles} +\graphicspath{{\subfix{../images/}}} + +\begin{document} + +\section{Equations of electrodynamics and classical mechanics}\label{sec:appendix_a} +Newton's second law +\begin{equation}\label{eq:newton_second} + m \ddot{\mathbf{r}} = \sum_{i} \mathbf{F}_{i} +\end{equation}% +% +Lorentz force +\begin{equation}\label{eq:lorentz_force} + \mathbf{F} = q \mathbf{E} + q \mathbf{v} \times \mathbf{B}, +\end{equation}% +% +Electric field +\begin{equation}\label{eq:e_field_potential} + \mathbf{E} = - \nabla V +\end{equation} % +% +Electric field at a point $\mathbf{r}$ +\begin{equation}\label{eq:e_field_point} + \mathbf{E} = k_{e} \sum_{j=1}^{n} q_j \frac{\mathbf{r} - \mathbf{r}_{j}}{|\mathbf{r} - \mathbf{r}_{j}|^{3}} +\end{equation} + + + +\end{document} \ No newline at end of file diff --git a/latex/appendix/appendix_b.tex b/latex/appendix/appendix_b.tex new file mode 100644 index 0000000..772f73e --- /dev/null +++ b/latex/appendix/appendix_b.tex @@ -0,0 +1,145 @@ +\documentclass[../main.tex]{subfiles} +\graphicspath{{\subfix{../images/}}} + +\begin{document} +\section{Derivation of equations}\label{sec:appendix_b} +% 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} + + +\end{document} + +% \begin{align*} +% \omega_{+} - \omega_{-} &= \frac{\omega_{0} + \sqrt{\omega_{0}^{2} - 2 \omega_{z}^{2}}}{2} - \frac{\omega_{0} - \sqrt{\omega_{0}^{2} - 2 \omega_{z}^{2}}}{2} \\ +% &= +% \end{align*} \ No newline at end of file diff --git a/latex/appendix/appendix_c.tex b/latex/appendix/appendix_c.tex new file mode 100644 index 0000000..0a3edd7 --- /dev/null +++ b/latex/appendix/appendix_c.tex @@ -0,0 +1,21 @@ +\documentclass[../main.tex]{subfiles} +\graphicspath{{\subfix{../images/}}} + +\begin{document} + +\section{Numbers and units}\label{sec:appendix_c} +\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} + +\end{document} \ No newline at end of file diff --git a/latex/images/3d_plot.pdf b/latex/images/3d_plot.pdf index 98a1825778ede984b5d8eed1a4c17008dd6329b8..4a25654cd8fb80fb1dfea02e85657ed6e4cff11f 100644 GIT binary patch delta 62421 zcmV(vKzsI0FUob8mN@-$FSvx0L|P@zyr5OZFSqi zUk17x@axli{Z>StA|+B3OSFk5QN+E;bNAUhcV^^@h!rdT^1uDt|KorAcYph@|N5`K z{+oaE?Js})_V51lxBB&W`u88c%GclN-~ahlzy4bP{=>JT-~RsFe^}-F(NDkRPkqeq zOTU%RPyg$;Z~y+=U;Zoo$$$PT-*4S(>@v^!N570)>Rn}@um8li{q=8r;M+g->2Kfm z`94SOwSCq0{Vv^DKka*)XX(>1^vkag^}4>Vz5AXT|MADaF`PHv^2Vp;pz8Vk<6q~< zw{Nx7?|qI|)>kdAr@!xeT@ttkPx3+xR_D?@<(In~(mG-sf_pwgt@zuZWe};b7+~|w{@?Ul4yMFzR z{{7uo)eFlt_YYR(A9f$#{*6}T-{>*wkJnh^!5I1WSHJr9m;ZhJs=Ar6>YZz~{q?JV z&peDd`>OMN{YClS+nl=g+4bum{PN>twe;`vjM3^`^?Y0Pd(pEteNc<=SAXw)^Q-^m 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zgAOMtqV1c@1YwR#SLn(5@Ojj}pWJ>x-2{Bwk2cda__aZ@kVKNTC{;M`e4BOH%lyPR zEeUiIUy8c8eI7gcM=#mf{T}eS@9w6aB**Q$!hB^bx{Lhf5Vum^mCu84!UWaR-!f)9Iis|C+nl*6;tREJsw6nhe$eq>anDiL|0DBh3pzD7vq$x1@21Bef@l%)vFdGS(5 z30;&r%9aH)GQ}xM%=xBL(1~(X=&IPID)JT?N*wuCQlkl}l&pW;gED7IGGn{{rBX@z z=2ceOxnTSQAMxb)Gx(2wohkR8%*k>Gf(VkDuZ2M*&37IVB%HVDg?S_~tnN~hm)rcA zQ*r4}^U7jz^LA@2)Wy%GnTC=#r_gb!s{1pOl2jwYkfSh zD29C~d0D=z&K#nsqaKMm~ipq<@{6B>H0m=XX literal 0 HcmV?d00001 diff --git a/latex/main.tex b/latex/main.tex index f9c63f7..e7ab021 100644 --- a/latex/main.tex +++ b/latex/main.tex @@ -10,6 +10,7 @@ % Allows special characters (including æøå) \usepackage[utf8]{inputenc} \usepackage{fontawesome} +% \usepackage{tabularx} % \usepackage[english]{babel} %% Note that you may need to download some of these packages manually, it depends on your setup. @@ -24,12 +25,12 @@ \usepackage{listings} % display code \usepackage{subfigure} % imports a lot of cool and useful figure commands % \usepackage{float} -% \usepackage[section]{placeins} +%\usepackage[section]{placeins} \usepackage{algorithm} \usepackage[noend]{algpseudocode} \usepackage{subfigure} \usepackage{tikz} -% \usetikzlibrary{quantikz2} +\usetikzlibrary{quantikz2} % defines the color of hyperref objects % Blending two colors: blue!80!black = 80% blue and 20% black \hypersetup{ % this is just my personal choice, feel free to change things @@ -63,9 +64,6 @@ % Introduction \subfile{sections/introduction} -% Theory -\subfile{sections/theory} - % Methods \subfile{sections/methods} @@ -76,7 +74,10 @@ \subfile{sections/conclusion} -\subfile{appendix/appendix} +\appendix +\subfile{appendix/appendix_a} +\subfile{appendix/appendix_b} +\subfile{appendix/appendix_c} \onecolumngrid diff --git a/latex/sections/introduction.tex b/latex/sections/introduction.tex index 567af9f..0ef8ecb 100644 --- a/latex/sections/introduction.tex +++ b/latex/sections/introduction.tex @@ -2,12 +2,12 @@ \graphicspath{{\subfix{../images/}}} \begin{document} - +% \section{Introduction} We are surrounded by matter, which are made up of elementary particles. In the field of physics we want to understand the properties of these particles, measure their physical quantities, not to mention explain the origin of mass \cite{britannica:2023:matter}. However, to study a particle, it is necessary isolate and contain it over time. The Penning trap is a device, able to confine charged particles for a period of time. This concept was evolved from F. M. Penning's implementation of magnetic fiels to a vaccum gauge, and J. R. Pierce's work with electron beams, and put into practice by Hans Dehmelt. In 1973 Dehmelt and his group of researchers were able to 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. - +% \end{document} \ No newline at end of file diff --git a/latex/sections/methods.tex b/latex/sections/methods.tex index 125bd37..283e16d 100644 --- a/latex/sections/methods.tex +++ b/latex/sections/methods.tex @@ -2,40 +2,101 @@ \graphicspath{{\subfix{../images/}}} \begin{document} -\section{Methods} +\section{Theory and methods} +% Problem 1 +When we study the Penning traps effect on a particle with a charge $q$, we need to consider the forces acting on the particle. We can use Newton's second law \eqref{eq:newton_second} to determine the position of a particle as a function of time. In addition, we introduce the Lorentz force \eqref{eq:lorentz_force}, which describes the force on the particle. The position can be described by % +% +\begin{equation}\label{eq:newton_lorentz} + 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 +\begin{align} + \label{eq:motion_x} + \ddot{x} - \omega_{0} \dot{y} - \frac{1}{2} \omega_{z}^{2} x &= 0, \\ + \label{eq:motion_y} + \ddot{y} - \omega_{0} \dot{x} - \frac{1}{2} \omega_{z}^{2} y &= 0, \\ + \label{eq:motion_z} + \ddot{z} + \omega_{z}^{2} z &= 0, +\end{align} % +% +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 $q > 0$. + +% 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 +\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 +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*} + \omega_{\pm} = \frac{\omega_{0} \pm \sqrt{\omega_{0}^{2} - 2 \omega_{z}^{2}}}{2}. +\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 +\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_{-}) +\end{align} % +% Problem 4 +However, to obtain a bound on the particle's movement in the radial direction, we have to ensure that +\begin{align*} + |f(t)| &= \sqrt{(x(t))^{2} + (y(t))^{2}}. \\ +\end{align*} % +When $t \rightarrow \infty$, the upper and lower limits are +\begin{align} + R_{+} &= A_{+} + A_{-} \\ + R_{-} &= |A_{+} - A_{-}|, +\end{align} +derived in sec. \ref{sec:bounded_solution}. + +% Problem 3 +To obtain the bounded solution, we need to consider the expression +\begin{equation*} + \alpha = (\omega_{+} - \omega_{-}) t +( \phi_{+} - \phi_{-}). +\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. +\begin{align*} + \omega_{0}^{2} - 2 \omega_{z}^{2} = 0, \quad \omega_{0} > 2 \omega_{z} +\end{align*} +\begin{equation}\label{eq:angular_rate} + \omega_{\pm} = \frac{\omega_{0} \pm \sqrt{\omega_{0}^{2} - 2 \omega_{z}^{2}}}{2} +\end{equation} + +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 + + + -\subsection{Units and constants} -Before continuing, we need to define the units we'll be working with. -Since we are working with particles, we need small units to work with so the -numbers we are working with aren't so small that they could potentially lead -to large round-off errors in our simulation. The units that we will use are listed in Table~\ref{tab:units}. \begin{table}[H] - \begin{center} + \centering \begin{tabular}[c]{lll} - Dimension & Unit & Symbol \\ + Constant & Value & \\ \hline - Length & micrometer & $\mu m$ \\ - Time & microseconds & $\mu s$ \\ - Mass & atomic mass unit & $u$ \\ - Charge & the elementary charge & $e$ \\ + $B_{0}$ & $1.00 T$ & $9.65 \cross 10^{1} \frac{u}{(\mu s) e}$ \\ + $V_{0}$ & $25.0 mV$ & $2.41 \cross 10^{6} \frac{u (\mu m)^{2}}{(\mu s)^{2} e}$ \\ + $d$ & $500 \ \mu m$ & \\ \hline \end{tabular} - \end{center} - \caption{The set of units we'll be working with.}\label{tab:units} + \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} -With these base units, we get -\begin{equation} - k_e = 1.3893533 \cdot 10^5 \frac{u(\mu m)^3}{(\mu s)^2 e}, -\end{equation} -and we get that the unit for magnetic field strength (Tesla, $T$) and electric potential (Volt, $V$) are -\begin{equation} - \begin{split} - T &= 9.64852558 \cdot 10^1 \frac{u}{(\mu s) e} \\ - V &= 9.64852558 \cdot 10^7 \frac{u (\mu m)^2}{(\mu s)^2 e}. \\ - \end{split} - \label{eq:} -\end{equation} \subsection{Dealing with a multi--particle system} @@ -230,7 +291,6 @@ and would reduce the required amount of loops down to 4. \subsection{Relative error and error convergance rate} \subsection{Tools} -We used matplotlib +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. -%\biblio \end{document} diff --git a/latex/sections/results.tex b/latex/sections/results.tex index c48a6b3..9437367 100644 --- a/latex/sections/results.tex +++ b/latex/sections/results.tex @@ -4,7 +4,86 @@ \begin{document} \section{Results and Discussion} -From equation \eqref{eq:general_solution} we can find +% 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}. -Problem 9: we have to deal with $|\mathbf{r}| = d$ +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} +\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.} + \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] + \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} +\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} + \centering + \includegraphics[width=\linewidth]{images/phase_space_2_particles_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_2_particles_z.pdf} + \caption{Phase space plot of two particles in z-direction.} + \label{fig:phase_space_2z} +\end{figure} + +\begin{figure} + \centering + \includegraphics[width=\linewidth]{images/3d_plot.pdf} + \caption{3D plot of particles-} + \label{fig:3d_particles} +\end{figure} + +\begin{figure} + \centering + \includegraphics[width=\linewidth]{images/phase_space_2_particles.pdf} + \caption{Phase space plot of two particles.} + \label{fig:phase_space_2} +\end{figure} + + +\begin{figure} + \centering + \includegraphics[width=\linewidth]{images/particles_left.pdf} + \caption{Fraction of particles left in the Penning trap, with a given amplitude $f$.} + \label{fig:particles_left} +\end{figure} \end{document} \ No newline at end of file From 872b84d0d1cc9be2664a4899b237a381e7fc7b70 Mon Sep 17 00:00:00 2001 From: Janita Willumsen Date: Sun, 22 Oct 2023 17:15:13 +0200 Subject: [PATCH 11/15] Change labels --- src/scripts/plot_phase_space.py | 4 ++-- src/scripts/plot_relative_error.py | 16 ++++++++-------- 2 files changed, 10 insertions(+), 10 deletions(-) diff --git a/src/scripts/plot_phase_space.py b/src/scripts/plot_phase_space.py index b989961..48326e5 100644 --- a/src/scripts/plot_phase_space.py +++ b/src/scripts/plot_phase_space.py @@ -62,9 +62,9 @@ def main(): axs2[1].set(xlabel=r"t $(\mu s)$", ylabel = r"z $(\mu m)$") axs2[0].set(ylabel = r"z $(\mu m)$") - axs1[i].legend() + axs1[i].legend(loc="upper right") # axs1[i].set_title(title) - axs2[i].legend() + axs2[i].legend(loc="upper right") # axs2[i].set_title(title) fig1.savefig("../latex/images/phase_space_2_particles_x.pdf", bbox_inches="tight") diff --git a/src/scripts/plot_relative_error.py b/src/scripts/plot_relative_error.py index 10106af..1adcb72 100644 --- a/src/scripts/plot_relative_error.py +++ b/src/scripts/plot_relative_error.py @@ -27,10 +27,10 @@ def main(): "32000_steps.txt", ] labels = [ - r"4000 steps", - r"8000 steps", - r"16000 steps", - r"32000 steps", + r"$n_1$", + r"$n_2$", + r"$n_3$", + r"$n_4$", ] titles = [ "Relative error for the RK4 method", @@ -47,17 +47,17 @@ def main(): max_err.append(max(r)) axs1[i].plot(t, r, label=label) - axs1[i].set(ylabel = r"relative_error $(\mu m)$") # xlabel=r"t $(\mu s)$", + axs1[i].set(ylabel = r"Relative error $(\mu m)$") # xlabel=r"t $(\mu s)$", axs1[1].set_xlabel(r"t $(\mu s)$") - axs1[i].legend() + axs1[i].legend(loc="upper right") # axs1[i].set_title(title) conv_rate = 1/3 * sum([np.log10(max_err[i+1]/max_err[i])/np.log10(.5) for i in range(3)]) print(conv_rate) - # fig1.savefig("../latex/images/phase_space_2_particles_x.pdf") - plt.show() + fig1.savefig("../latex/images/relative_error.pdf") + # plt.show() if __name__ == "__main__": From 4e14915b04ee473d37fa8b321f6a4759ca4d9014 Mon Sep 17 00:00:00 2001 From: Janita Willumsen Date: Sun, 22 Oct 2023 17:16:47 +0200 Subject: [PATCH 12/15] Stuff --- src/main.cpp | 6 +++--- 1 file changed, 3 insertions(+), 3 deletions(-) diff --git a/src/main.cpp b/src/main.cpp index 19ef8fd..e98eaa0 100644 --- a/src/main.cpp +++ b/src/main.cpp @@ -176,11 +176,11 @@ int main() simulate_two_particles(); - simulate_single_particle_with_different_steps(); + simulate_single_particle_with_different_steps(); - // simulate_100_particles(); + simulate_100_particles(); - // simulate_100_particles_with_time_potential(); + simulate_100_particles_with_time_potential(); double end = omp_get_wtime(); From b985a2c25fc00c607343cb46b991c4b21debee7d Mon Sep 17 00:00:00 2001 From: Cory Date: Sun, 22 Oct 2023 17:17:28 +0200 Subject: [PATCH 13/15] Changes --- src/scripts/plot_particles_left.py | 63 ++++++++++++++++++------------ src/scripts/plot_phase_space.py | 14 +++---- 2 files changed, 44 insertions(+), 33 deletions(-) diff --git a/src/scripts/plot_particles_left.py b/src/scripts/plot_particles_left.py index 9884cd2..0f47985 100644 --- a/src/scripts/plot_particles_left.py +++ b/src/scripts/plot_particles_left.py @@ -23,35 +23,46 @@ def main(): "mediumaquamarine", "mediumseagreen" ] - with open("output/time_dependent_potential/res.txt") as f: - lines = f.readlines() - x = [] - y1 = [] - y2 = [] - y3 = [] - for line in lines: - l = line.strip().split(",") - x.append(float(l[0])) - y1.append(float(l[1])) - y2.append(float(l[2])) - y3.append(float(l[3])) + files = [ + "output/time_dependent_potential/wide_sweep.txt", + "output/time_dependent_potential/narrow_sweep.txt", + "output/time_dependent_potential/narrow_sweep_interactions.txt", + ] + outputs = [ + "particles_left_wide_sweep.pdf", + "particles_left_narrow_sweep.pdf", + "particles_left_narrow_sweep_interactions.pdf", + ] + for file, output in zip(files, outputs): + with open(file) as f: + lines = f.readlines() + x = [] + y1 = [] + y2 = [] + y3 = [] + for line in lines: + l = line.strip().split(",") + x.append(float(l[0])) + y1.append(float(l[1])) + y2.append(float(l[2])) + y3.append(float(l[3])) - fig, ax = plt.subplots() - ax.plot(x, y1, label=r"$f_{1} = 0.1$", color=colors[0]) - ax.plot(x, y2, label=r"$f_{2} = 0.4$", color=colors[2]) - ax.plot(x, y3, label=r"$f_{3} = 0.7$", color=colors[4]) + fig, ax = plt.subplots() + ax.plot(x, y1, label=r"$f_{1} = 0.1$", color=colors[0]) + ax.plot(x, y2, label=r"$f_{2} = 0.4$", color=colors[2]) + ax.plot(x, y3, label=r"$f_{3} = 0.7$", color=colors[4]) - ax.set_xlabel(r"Frequency $\omega_V$ (MHz)") - ax.set_xlim((0, 2.8)) - ax.set_ylabel(r"Fraction of particles left") - ax.set_ylim((-0.1, 1.1)) - # plt.title(r"The fraction of particles left in the Penning trap " - # "after 500 microseconds for different amplitudes and frequencies") - - ax.legend(loc="upper right") + ax.set_xlabel(r"Frequency $\omega_V$ (MHz)") + # ax.set_xlim((0, 2.8)) + ax.set_ylabel(r"Fraction of particles left") + # ax.set_ylim((-0.1, 1.1)) + # plt.title(r"The fraction of particles left in the Penning trap " + # "after 500 microseconds for different amplitudes and frequencies") + + ax.legend(loc="upper right") - # plt.show() - fig.savefig("../latex/images/particles_left.pdf", bbox_inches="tight") + # plt.show() + fig.savefig(f"../latex/images/{output}", bbox_inches="tight") if __name__ == "__main__": main() diff --git a/src/scripts/plot_phase_space.py b/src/scripts/plot_phase_space.py index b989961..af17c7c 100644 --- a/src/scripts/plot_phase_space.py +++ b/src/scripts/plot_phase_space.py @@ -42,19 +42,19 @@ def main(): fig2, axs2 = plt.subplots(2,1,sharex=True) for i, (file, label) in enumerate(zip(files, labels)): for j, (dir, linestyle) in enumerate(zip(directories, linestyles)): + r = [] + v = [] with open(dir+file+"_r.txt") as f: lines = f.readlines() - t = np.linspace(0, 50, len(lines)) r = np.array([list(map(float, line.strip().split(","))) for line in lines]) - axs1[i].plot(t, r[:,0], label=f"{label[j]} r", linestyle=linestyle) - axs2[i].plot(t, r[:,2], label=f"{label[j]} r", linestyle=linestyle) with open(dir+file+"_v.txt") as f: lines = f.readlines() - t = np.linspace(0, 50, len(lines)) - r = np.array([list(map(float, line.strip().split(","))) for line in lines]) - axs1[i].plot(t, r[:,0], label=f"{label[j]} v", linestyle=linestyle) - axs2[i].plot(t, r[:,2], label=f"{label[j]} v", linestyle=linestyle) + v = np.array([list(map(float, line.strip().split(","))) for line in lines]) + axs1[i].plot(r[:,0], v[:,0], label=label[j]) + axs2[i].plot(r[:,2], v[:,2], label=label[j]) + axs1[i].axis("equal") + axs2[i].axis("equal") axs1[1].set(xlabel=r"t $(\mu s)$", ylabel = r"x $(\mu m)$") From 122470031288167f5e2a010b55ba510814be0f28 Mon Sep 17 00:00:00 2001 From: Janita Willumsen Date: Sun, 22 Oct 2023 17:24:06 +0200 Subject: [PATCH 14/15] Update references --- latex/main.pdf | Bin 673380 -> 673380 bytes latex/references/references.bib | 141 ++++++++++++++++++-------------- latex/sections/methods.pdf | Bin 0 -> 255084 bytes 3 files changed, 79 insertions(+), 62 deletions(-) create mode 100644 latex/sections/methods.pdf diff --git a/latex/main.pdf b/latex/main.pdf index aee475488d41385e4720c5be90d24f22f6d4462a..4174abc41b2f95134343ac79d5fc6358b81378e2 100644 GIT binary patch delta 124 zcmaFTqxqyqv!R8tg{g(Pg{6hHg{_6Xg`=4P4yJOEltcU9nH)w%nX6R#nev0hLDo&d`?_pEC6iQA2k2~ diff --git a/latex/references/references.bib b/latex/references/references.bib index 6a49b46..1ada635 100644 --- a/latex/references/references.bib +++ b/latex/references/references.bib @@ -1,87 +1,104 @@ @book{vogel:2018, - author = {Manuel Vogel}, - title = {Particle Confinement in Penning Traps}, - subtitle = {An Introduction}, - publisher = {Springer Cham}, - year = {2018}, - edition = {1}, - abstract = {An introduction to the field of Penning traps and related experimental techniques. The book is motivated by the observation that often a vast number of different resources have to be explored to gain a good overview of Penning trap principles. This is especially true for students who experience additional difficulty due to the different styles of presentation and notation.} + author = {Manuel Vogel}, + title = {Particle Confinement in Penning Traps}, + subtitle = {An Introduction}, + publisher = {Springer Cham}, + year = {2018}, + edition = {1}, + abstract = {An introduction to the field of Penning traps and related experimental techniques. The book is motivated by the observation that often a vast number of different resources have to be explored to gain a good overview of Penning trap principles. This is especially true for students who experience additional difficulty due to the different styles of presentation and notation.} } @inbook{vogel:2018:ch5, - author = {Manuel Vogel}, - title = {Particle Confinement in Penning Traps}, - subtitle = {An Introduction}, - publisher = {Springer Cham}, - year = {2018}, - edition = {1}, - pages = {45--50} + author = {Manuel Vogel}, + title = {Particle Confinement in Penning Traps}, + subtitle = {An Introduction}, + publisher = {Springer Cham}, + year = {2018}, + edition = {1}, + pages = {45--50} } @inbook{vogel:2018:ch1, - author = {Manuel Vogel}, - title = {Particle Confinement in Penning Traps}, - subtitle = {An Introduction}, - publisher = {Springer Cham}, - year = {2018}, - edition = {1--6} + author = {Manuel Vogel}, + title = {Particle Confinement in Penning Traps}, + subtitle = {An Introduction}, + publisher = {Springer Cham}, + year = {2018}, + edition ={1--6} } -@article{hunter:2007, - author = {Hunter, J. D.}, - title = {Matplotlib: A 2D graphics environment}, - journal = {Computing in Science \& Engineering}, - volume = {9}, - number = {3}, - pages = {90--95}, - abstract = {Matplotlib is a 2D graphics package used for Python for - application development, interactive scripting, and publication-quality - image generation across user interfaces and operating systems.}, +@article{hunter:2007:matplotlib, + author = {Hunter, J. D.}, + title = {Matplotlib: A 2D graphics environment}, + journal = {Computing in Science \& Engineering}, + volume = {9}, + number = {3}, + pages = {90--95}, + abstract = {Matplotlib is a 2D graphics package used for Python for + application development, interactive scripting, and publication-quality + image generation across user interfaces and operating systems.}, publisher = {IEEE COMPUTER SOC}, - doi = {10.1109/MCSE.2007.55}, - year = 2007 + doi = {10.1109/MCSE.2007.55}, + year = 2007 } @inbook{lindstrom:2016:ch3, - author = {Tom Lindstrøm}, - title = {Kalkulus}, - publisher = {Universitetsforlaget}, - year = {2016}, - edition = {4}, - pages = {133--137} + author = {Tom Lindstrøm}, + title = {Kalkulus}, + publisher = {Universitetsforlaget}, + year = {2016}, + edition = {4}, + pages = {133--137} } @inbook{lindstrom:2016:ch10:5, - author = {Tom Lindstrøm}, - title = {Kalkulus}, - publisher = {Universitetsforlaget}, - year = {2016}, - edition = {4}, - pages = {589--606} + author = {Tom Lindstrøm}, + title = {Kalkulus}, + publisher = {Universitetsforlaget}, + year = {2016}, + edition = {4}, + pages = {589--606} } -@article{britannica:2023:matter, - author = {The Editors of Encyclopaedia Britannica}, - title = {matter}, - year = {2023}, - url = {https://www.britannica.com/science/matter}, - urldate = {2023-10-17} +@online{britannica:2023:matter, + author = {The Editors of Encyclopaedia Britannica}, + title = {matter}, + year = {2023}, + url = {https://www.britannica.com/science/matter}, + urldate = {2023-10-17} } @inbook{gjevik:2014:appendix, - author = {Bjørn Gjevik and Morten Wang Fagerland}, - title = {Feltteori og vektoranalyse}, - publisher = {Farleia Forlag}, - year = {2018}, - pages = {162--163} + author = {Bjørn Gjevik and Morten Wang Fagerland}, + title = {Feltteori og vektoranalyse}, + publisher = {Farleia Forlag}, + year = {2018}, + pages = {162--163} } +@article{waskom:2021:seaborn, + author = {Michael L. Waskom}, + title = {seaborn: statistical data visualization}, + journal = {Journal of Open Source Software}, + volume = {6}, + number = {60}, + publisher = {The Open Journal}, + year = {2021}, + doi = {10.21105/joss.03021}, + url = {https://doi.org/10.21105/joss.03021}, + pages = {3021} + } -@article{rk4_method, - author = "Morten Hjorth-Jensen", - title = "Computational Physics, Lecture Notes Fall 2015", - journal = "Department of Physics, University of Oslo", - year = "2015", - chapter = "8.4", - pages = "250--252", +@online{openmp:2018, + author = {OpenMP}, + title = {OpenMP Application Programming Interface}, + year = {2018}, + url = {https://www.openmp.org/wp-content/uploads/OpenMP-API-Specification-5.0.pdf}, + urldate = {2023-10-22} } + +@misc{penning:fig, + key = , + url = {https://www.med.physik.uni-muenchen.de/research/nuclear-science/nuclear-masses/mlltrap/layout/traps/index.html} + note = {Configuration figure a} +} \ No newline at end of file diff --git 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