{
"cells": [
{
"cell_type": "markdown",
"id": "coordinate-balance",
"metadata": {},
"source": [
"# Particle motion in electromagnetic fields\n",
"\n",
"[A. Petrenko](https://www.inp.nsk.su/~petrenko/) (Novosibirsk, 2021)"
]
},
{
"cell_type": "markdown",
"id": "advisory-relations",
"metadata": {},
"source": [
"The equations of particle motion under the Lorentz force (in SI units)\n",
"\n",
"\\begin{equation}\n",
"\\frac{d \\mathbf{p}}{dt} = q \\mathbf{E} + q[\\mathbf{v} \\times \\mathbf{B}],\n",
"\\end{equation}\n",
"\n",
"\\begin{equation}\n",
"\\frac{d \\mathbf{r}}{dt} = \\mathbf{v},\n",
"\\end{equation}\n",
"\n",
"\\begin{equation}\n",
"\\mathbf{p} = \\gamma m \\mathbf{v}.\n",
"\\end{equation}\n",
"\n",
"Particle velocity $\\mathbf{v}$ can be derived from the momentum $\\mathbf{p}$ using the definition of relativistic factor $\\gamma = \\left(1 - \\frac{v^2}{c^2}\\right)^{-1/2} = \\sqrt{1 + \\left(\\frac{p}{mc}\\right)^2}$."
]
},
{
"cell_type": "markdown",
"id": "english-algeria",
"metadata": {},
"source": [
"Therefore the equations of motion\n",
"\n",
"\\begin{equation}\n",
"\\frac{ d\\mathbf{r} }{dt} = \\frac{\\mathbf{p}}{\\gamma m},\\\\\n",
"\\frac{d\\mathbf{p}}{dt} = q \\mathbf{E} + q\\left[\\frac{\\mathbf{p}}{\\gamma m} \\times \\mathbf{B}\\right].\n",
"\\end{equation}\n",
"\n",
"And the final 6d vector equation to solve $d\\mathbf{Y}/dt = \\mathbf{F}(t,\\mathbf{Y})$:"
]
},
{
"cell_type": "markdown",
"id": "finnish-coverage",
"metadata": {},
"source": [
"\\begin{equation}\n",
"\\frac{d}{dt}\n",
"\\left(\n",
"\\begin{array}{c}\n",
"x \\\\\n",
"y \\\\\n",
"z \\\\\n",
"p_x \\\\\n",
"p_y \\\\\n",
"p_z\n",
"\\end{array}\n",
"\\right) =\n",
"\\left(\n",
"\\begin{array}{c}\n",
"p_x/\\gamma m \\\\\n",
"p_y/\\gamma m \\\\\n",
"p_z/\\gamma m \\\\\n",
"qE_x + q(p_y B_z - p_z B_y)/\\gamma m \\\\\n",
"qE_y - q(p_x B_z - p_z B_x)/\\gamma m \\\\\n",
"qE_z + q(p_x B_y - p_y B_x)/\\gamma m\n",
"\\end{array}\n",
"\\right).\n",
"\\end{equation}"
]
},
{
"cell_type": "code",
"execution_count": 1,
"id": "constitutional-medium",
"metadata": {},
"outputs": [],
"source": [
"import numpy as np\n",
"from scipy.integrate import solve_ivp"
]
},
{
"cell_type": "markdown",
"id": "happy-overall",
"metadata": {},
"source": [
"Define external fields:"
]
},
{
"cell_type": "code",
"execution_count": 2,
"id": "premium-swedish",
"metadata": {},
"outputs": [],
"source": [
"def E(x,y,z,t): # V/m\n",
" Ex = 0; Ey = 0; Ez = 0\n",
" return Ex, Ey, Ez"
]
},
{
"cell_type": "code",
"execution_count": 3,
"id": "genuine-tanzania",
"metadata": {},
"outputs": [],
"source": [
"def B(x,y,z,t): # T\n",
" Bx = 0; By = 0; Bz = 1.0 # T\n",
" return Bx, By, Bz"
]
},
{
"cell_type": "markdown",
"id": "three-politics",
"metadata": {},
"source": [
"Define some constants:"
]
},
{
"cell_type": "code",
"execution_count": 4,
"id": "junior-minneapolis",
"metadata": {},
"outputs": [],
"source": [
"c = 299792458 # m/sec\n",
"e = 1.602176634e-19 # C\n",
"eV = e # J\n",
"keV = 1e3*eV; MeV = 1e6*eV; GeV = 1e9*eV\n",
"\n",
"sec = 1; ns = 1e-9*sec\n",
"\n",
"# electron:\n",
"q = -e\n",
"mc2 = 511*keV # J\n",
"\n",
"mc = mc2/c # kg*m/sec\n",
"m = mc/c # kg"
]
},
{
"cell_type": "code",
"execution_count": 5,
"id": "indian-output",
"metadata": {},
"outputs": [],
"source": [
"def F(t,Y):\n",
" x = Y[0]; y = Y[1]; z = Y[2]\n",
" px = Y[3]; py = Y[4]; pz = Y[5]\n",
" gamma = np.sqrt(1 + (px*px + py*py + pz*pz)/(mc*mc))\n",
" Ex, Ey, Ez = E(x,y,z,t) # V/m\n",
" Bx, By, Bz = B(x,y,z,t) # T\n",
" \n",
" return np.array([\n",
" px/(gamma*m),\n",
" py/(gamma*m),\n",
" pz/(gamma*m),\n",
" q*Ex + q*(py*Bz-pz*By)/(gamma*m),\n",
" q*Ey - q*(px*Bz-pz*Bx)/(gamma*m),\n",
" q*Ez + q*(px*By-py*Bx)/(gamma*m)\n",
" ])"
]
},
{
"cell_type": "markdown",
"id": "interpreted-crash",
"metadata": {},
"source": [
"Set the initial particle coordinates"
]
},
{
"cell_type": "code",
"execution_count": 6,
"id": "maritime-right",
"metadata": {},
"outputs": [],
"source": [
"x = 0; y = 0; z = 0\n",
"px = 0.5*GeV/c; py = 0; pz = 0.1*GeV/c\n",
"\n",
"Y0 = np.array([x,y,z,px,py,pz])"
]
},
{
"cell_type": "code",
"execution_count": 7,
"id": "bibliographic-radiation",
"metadata": {},
"outputs": [],
"source": [
"t_span = (0, 200*ns) # sec\n",
"\n",
"t_eval = np.linspace(t_span[0], t_span[1], 500) # points of output"
]
},
{
"cell_type": "code",
"execution_count": 8,
"id": "combined-badge",
"metadata": {},
"outputs": [],
"source": [
"#help(solve_ivp)"
]
},
{
"cell_type": "code",
"execution_count": 9,
"id": "spread-renaissance",
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"CPU times: user 20.7 ms, sys: 2.45 ms, total: 23.2 ms\n",
"Wall time: 24.1 ms\n"
]
}
],
"source": [
"%%time\n",
"sol = solve_ivp(F, t_span, Y0, t_eval=t_eval, max_step=5*ns) # rtol=1e-4"
]
},
{
"cell_type": "markdown",
"id": "detected-juvenile",
"metadata": {},
"source": [
"## Plot the results"
]
},
{
"cell_type": "code",
"execution_count": 10,
"id": "separate-sector",
"metadata": {},
"outputs": [],
"source": [
"import plotly.io as pio\n",
"import plotly.graph_objects as go"
]
},
{
"cell_type": "code",
"execution_count": 11,
"id": "19628171-50a8-4593-9e73-117b308e5787",
"metadata": {},
"outputs": [],
"source": [
"#pio.renderers"
]
},
{
"cell_type": "code",
"execution_count": 12,
"id": "e304e5d7-07f7-4509-94ef-485f0b713cd4",
"metadata": {},
"outputs": [],
"source": [
"pio.renderers.default = \"plotly_mimetype+notebook\" # both notebook & html from nbconvert are interactive\n",
"#pio.renderers.default = \"svg\" # svg figures (for publications)"
]
},
{
"cell_type": "code",
"execution_count": 13,
"id": "5713334f-285e-45d4-8e5a-d778aace4674",
"metadata": {},
"outputs": [],
"source": [
"pio.templates.default = pio.templates[\"simple_white\"]"
]
},
{
"cell_type": "code",
"execution_count": 14,
"id": "protecting-arnold",
"metadata": {},
"outputs": [],
"source": [
"x, y, z, px, py, pz = sol.y"
]
},
{
"cell_type": "code",
"execution_count": 15,
"id": "blind-louisiana",
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
" \n",
" "
]
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"output_type": "display_data"
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"source": [
"fig = go.Figure()\n",
"\n",
"fig.add_trace( go.Scatter(x=t_eval/ns, y=y, mode='lines', line_width=3, line_color=\"blue\"))\n",
"\n",
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},
{
"cell_type": "code",
"execution_count": 16,
"id": "former-shower",
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"fig = go.Figure()\n",
"\n",
"fig.add_trace( go.Scatter(x=x, y=y, mode='lines', line_width=3, line_color=\"blue\"))\n",
"\n",
"fig.update_xaxes(title_text=\"x (m)\")\n",
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{
"cell_type": "markdown",
"id": "arctic-mauritius",
"metadata": {},
"source": [
"Check if the [Larmor radius](https://en.wikipedia.org/wiki/Gyroradius) agrees with the theory:"
]
},
{
"cell_type": "code",
"execution_count": 17,
"id": "statewide-hands",
"metadata": {},
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},
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"Bx, By, Bz = B(0,0,0,0) # T\n",
"\n",
"R = px[0]/(q*Bz) # m\n",
"R"
]
},
{
"cell_type": "code",
"execution_count": 18,
"id": "great-parent",
"metadata": {},
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""
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"source": [
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},
{
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"id": "timely-cycle",
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"'/data/shared/Acc_Course_2025/2025.02.20_General_Tracking'"
]
},
"execution_count": 19,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"pwd"
]
},
{
"cell_type": "code",
"execution_count": 20,
"id": "612d8953-d318-42a2-a2a0-506b9895686d",
"metadata": {},
"outputs": [],
"source": [
"%load_ext watermark"
]
},
{
"cell_type": "code",
"execution_count": 21,
"id": "applicable-dependence",
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"Python implementation: CPython\n",
"Python version : 3.12.2\n",
"IPython version : 8.27.0\n",
"\n",
"conda environment: n/a\n",
"\n",
"Compiler : GCC 12.3.0\n",
"OS : Linux\n",
"Release : 6.11.0-19-generic\n",
"Machine : x86_64\n",
"Processor : x86_64\n",
"CPU cores : 20\n",
"Architecture: 64bit\n",
"\n",
"plotly : 5.24.1\n",
"numpy : 1.26.4\n",
"scipy : 1.13.1\n",
"sys : 3.12.2 | packaged by conda-forge | (main, Feb 16 2024, 20:50:58) [GCC 12.3.0]\n",
"json : 2.0.9\n",
"IPython: 8.27.0\n",
"\n"
]
}
],
"source": [
"%watermark --iversions --conda --machine --python"
]
},
{
"cell_type": "code",
"execution_count": 23,
"id": "psychological-garage",
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"[NbConvertApp] Converting notebook general_tracking.ipynb to HTML\n",
"[NbConvertApp] Writing 4954973 bytes to general_tracking.html\n"
]
}
],
"source": [
"!jupyter nbconvert --to HTML general_tracking.ipynb"
]
},
{
"cell_type": "code",
"execution_count": null,
"id": "5e2c7092-14e2-48d1-a013-c17b061482b5",
"metadata": {},
"outputs": [],
"source": []
}
],
"metadata": {
"kernelspec": {
"display_name": "Python 3 (ipykernel)",
"language": "python",
"name": "python3"
},
"language_info": {
"codemirror_mode": {
"name": "ipython",
"version": 3
},
"file_extension": ".py",
"mimetype": "text/x-python",
"name": "python",
"nbconvert_exporter": "python",
"pygments_lexer": "ipython3",
"version": "3.12.2"
}
},
"nbformat": 4,
"nbformat_minor": 5
}