{ "cells": [ { "cell_type": "markdown", "metadata": { "editable": true, "slideshow": { "slide_type": "" }, "tags": [] }, "source": [ "# Demo\n", "## Vortex dynamics\n", "\n", "In this example, we are going to simulate vortex core dynamics. After creating a vortex structure, we are first going to displace it by applying an external magnetic field. We will then turn off the external field, and compute the time-development of the system, and then be able to see the dynamics of the vortex core.\n", "\n", "The sample is a two-dimensional Permalloy disk sample with $r=50 \\,\\text{nm}$ edge length and $10\\,\\text{nm}$ thickness. Its energy equation consists of ferromagnetic exchange, Zeeman, and demagnetisation energy terms:\n", " \n", "$$E = \\int_{V} \\left[-A\\mathbf{m}\\cdot\\nabla^{2}\\mathbf{m} - \\mu_{0}M_\\text{s}\\mathbf{m}\\cdot\\mathbf{H} + w_\\text{d}\\right] \\text{d}V,$$\n", "\n", "where $A = 13 \\,\\text{pJ}\\,\\text{m}^{-1}$ is the exchange energy constant, $M_\\text{s} = 8 \\times 10^{5} \\,\\text{A}\\,\\text{m}^{-1}$ magnetisation saturation, $w_\\text{d}$ demagnetisation energy density, $\\mathbf{H}$ an external magnetic field, and $\\mathbf{m}=\\mathbf{M}/M_\\text{s}$ the normalised magnetisation field.\n", "\n", "The magnetisation dynamics is governed by the Landau-Lifshitz-Gilbert equation consisting of precession and damping terms:\n", "\n", "$$\\frac{\\partial\\mathbf{m}}{\\partial t} = -\\frac{\\gamma_{0}}{1+\\alpha^{2}}\\mathbf{m}\\times\\mathbf{H}_\\text{eff} - \\frac{\\gamma_{0}\\alpha}{1+\\alpha^{2}}\\mathbf{m}\\times(\\mathbf{m}\\times\\mathbf{H}_\\text{eff}),$$\n", "\n", "where $\\gamma_{0} = 2.211 \\times 10^{5} \\,\\text{m}\\,\\text{A}^{-1}\\,\\text{s}^{-1}$ and $\\alpha = 0.05$ is the Gilbert damping.\n", "\n", "The (initial) magnetisation field is a vortex state, whose magnetisation at each point $(x, y, z)$ in the sample can be represented as $(m_{x}, m_{y}, m_{z}) = (-cy, cx, 0.1)$, with $c = 10^{-9} \\text{m}^{-1}$." ] }, { "cell_type": "code", "execution_count": 1, "metadata": { "tags": [ "nbval-ignore-output" ] }, "outputs": [], "source": [ "# Some initial configurations\n", "%config InlineBackend.figure_formats = ['svg'] # output matplotlib plots as SVG\n", "import pandas as pd\n", "import matplotlib.pyplot as plt\n", "\n", "pd.options.display.max_rows = 5\n", "pd.options.display.float_format = \"{:,.2e}\".format" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## System initialisation\n", "\n", "The Ubermag code for defining the micromagnetic system is:" ] }, { "cell_type": "code", "execution_count": 2, "metadata": { "tags": [] }, "outputs": [], "source": [ "import discretisedfield as df\n", "import micromagneticmodel as mm\n", "\n", "# Geometry\n", "r = 50e-9 # Radius of the thin nano magnetic disk (m)\n", "thickness = 10e-9 # sample thickness (m)\n", "\n", "# Material (Permalloy) parameters\n", "Ms = 8e5 # saturation magnetisation (A/m)\n", "A = 13e-12 # exchange energy constant (J/m)\n", "\n", "# Dynamics (LLG equation) parameters\n", "gamma0 = mm.consts.gamma0 # gyromagnetic ratio (m/As)\n", "alpha = 0.05 # Gilbert damping\n", "\n", "system = mm.System(name=\"vortex_dynamics\")\n", "\n", "# Energy equation. We omit Zeeman energy term, because H=0.\n", "system.energy = mm.Exchange(A=A) + mm.Demag()\n", "\n", "# Dynamics equation\n", "system.dynamics = mm.Precession(gamma0=gamma0) + mm.Damping(alpha=alpha)\n", "\n", "\n", "# initial magnetisation state\n", "def m_init(point):\n", " x, y, _ = point\n", " c = 1e9 # (1/m)\n", " return (-c * y, c * x, 0.1)\n", "\n", "\n", "# Defining the geometry of the material as a circular disk\n", "def Ms_func(point):\n", " x, y, _ = point\n", " if x**2 + y**2 <= r**2:\n", " return Ms\n", " else:\n", " return 0\n", "\n", "\n", "# Sample's centre is placed at origin\n", "region = df.Region(p1=(-r, -r, -thickness / 2), p2=(r, r, thickness / 2))\n", "mesh = df.Mesh(region=region, cell=(5e-9, 5e-9, 10e-9))\n", "\n", "system.m = df.Field(mesh, nvdim=3, value=m_init, norm=Ms_func, valid=\"norm\")" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "The system object is now defined and we can investigate some of its properties:" ] }, { "cell_type": "code", "execution_count": 3, "metadata": { "tags": [] }, "outputs": [ { "data": { "text/latex": [ "$- A \\mathbf{m} \\cdot \\nabla^{2} \\mathbf{m}-\\frac{1}{2}\\mu_{0}M_\\text{s}\\mathbf{m} \\cdot \\mathbf{H}_\\text{d}$" ], "text/plain": [ "Exchange(A=1.3e-11) + Demag()" ] }, "execution_count": 3, "metadata": {}, "output_type": "execute_result" } ], "source": [ "system.energy" ] }, { "cell_type": "code", "execution_count": 4, "metadata": { "tags": [] }, "outputs": [ { "data": { "text/latex": [ "$-\\frac{\\gamma_{0}}{1 + \\alpha^{2}} \\mathbf{m} \\times \\mathbf{H}_\\text{eff}-\\frac{\\gamma_{0} \\alpha}{1 + \\alpha^{2}} \\mathbf{m} \\times (\\mathbf{m} \\times \\mathbf{H}_\\text{eff})$" ], "text/plain": [ "Precession(gamma0=221276.14872118403) + Damping(alpha=0.05)" ] }, "execution_count": 4, "metadata": {}, "output_type": "execute_result" } ], "source": [ "system.dynamics" ] }, { "cell_type": "code", "execution_count": 5, "metadata": { "tags": [ "nbval-ignore-output" ] }, "outputs": 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" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "system.m.orientation.sel(\"z\").mpl()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Energy minimisation\n", "To carry out micromagnetic simulation, we need to use a micromagnetic calulator. We are going to use OOMMF for this.\n", "We can now relax the system in the absence of external magnetic field using energy minimisation driver (`MinDriver`):" ] }, { "cell_type": "code", "execution_count": 6, "metadata": { "tags": [ "nbval-ignore-output" ] }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Running OOMMF (ExeOOMMFRunner)[2024/08/09 18:15]... 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" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "import oommfc as oc # Micromagnetic Calculator\n", "\n", "md = oc.MinDriver()\n", "md.drive(system)\n", "\n", "system.m.orientation.sel(\"z\").mpl()" ] }, { "cell_type": "markdown", "metadata": { "editable": true, "slideshow": { "slide_type": "" }, "tags": [] }, "source": [ "## Displacement with magnetic field\n", "\n", "Now, we have a relaxed vortex state, with its core at the centre of the sample. As the next step, we want to add an external magnetic field $H=3.4 \\times 10^{4}\\,\\text{Am}^{-1}$ in the positive $x$-direction to displace the vortex core. We do that by adding the Zeeman energy term to the energy equation:" ] }, { "cell_type": "code", "execution_count": 7, "metadata": { "tags": [ "nbval-ignore-output" ] }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Running OOMMF (ExeOOMMFRunner)[2024/08/09 18:15]... 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" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "H = (3.4e4, 0, 0) # an external magnetic field (A/m)\n", "\n", "system.energy += mm.Zeeman(H=H)\n", "\n", "md.drive(system)\n", "system.m.orientation.sel(\"z\").mpl()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Free relaxation\n", "\n", "The vortex core is now displaced in the positive $y$-direction. As the last step, we are going to turn off the external magnetic field and simulate dynamics using `TimeDriver`. We are going to run simulation for $20\\,\\text{ns}$ and save the magnetisation in $500$ steps." ] }, { "cell_type": "code", "execution_count": 8, "metadata": { "tags": [] }, "outputs": [ { "data": { "application/vnd.jupyter.widget-view+json": { "model_id": "9b2bae448a8841dfaedaeb5d1dc76300", "version_major": 2, "version_minor": 0 }, "text/plain": [ "Running OOMMF (ExeOOMMFRunner): 0%| | 0/500 files written [00:00]" ] }, "metadata": {}, "output_type": "display_data" }, { "name": "stdout", "output_type": "stream", "text": [ "Running OOMMF (ExeOOMMFRunner)[2024/08/09 18:15] took 36.6 s\n" ] } ], "source": [ "system.energy -= mm.Zeeman(H=H)\n", "\n", "td = oc.TimeDriver()\n", "td.drive(system, t=20e-9, n=500, verbose=2)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "The final magnetisation state shows that the vortex core has moved back to the sample's centre." ] }, { "cell_type": "code", "execution_count": 9, "metadata": { "tags": [ "nbval-ignore-output" ] }, "outputs": [ 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tmxmymzE
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" ], "text/plain": [ " t mx my mz E\n", "0 4.00e-11 4.18e-01 5.03e-02 1.82e-02 3.94e-18\n", "1 8.00e-11 4.27e-01 1.92e-01 1.07e-02 3.89e-18\n", "2 1.20e-10 4.26e-01 2.63e-01 3.90e-02 3.86e-18\n", "3 1.60e-10 3.05e-01 2.82e-01 7.84e-03 3.83e-18\n", "4 2.00e-10 2.57e-01 3.97e-01 2.66e-02 3.80e-18" ] }, "execution_count": 10, "metadata": {}, "output_type": "execute_result" } ], "source": [ "system.table.data[[\"t\", \"mx\", \"my\", \"mz\", \"E\"]].head()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "We can now plot the average $m_{x}$, $m_{y}$ and $m_{z}$ values as taken from the table as a function of time to give us an idea of the vortex core position." ] }, { "cell_type": "code", "execution_count": 11, "metadata": { "tags": [ "nbval-ignore-output" ] }, "outputs": [ { "data": { "image/svg+xml": [ "\n", "\n", "\n", " \n", " \n", " \n", " \n", " 2024-08-09T18:16:01.263503\n", " image/svg+xml\n", " \n", " \n", " Matplotlib v3.9.1, https://matplotlib.org/\n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", "\n" ], "text/plain": [ "
" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "system.table.mpl(y=[\"mx\", \"my\", \"mz\"])" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### Spatially resolved data\n", "Finally, we are going to have a look at the magnetisation field at different time-steps using `micromagneticdata`." ] }, { "cell_type": "code", "execution_count": 12, "metadata": { "tags": [ "nbval-ignore-output" ] }, "outputs": [ { "data": { "text/html": [ "
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drive_numberdatetimedriveradaptern_threadstn
002024-08-0918:15:22MinDriveroommfcNoneNaNNaN
112024-08-0918:15:23MinDriveroommfcNoneNaNNaN
222024-08-0918:15:23TimeDriveroommfcNone2.00e-085.00e+02
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" ], "text/plain": [ " drive_number date time driver adapter n_threads t \\\n", "0 0 2024-08-09 18:15:22 MinDriver oommfc None NaN \n", "1 1 2024-08-09 18:15:23 MinDriver oommfc None NaN \n", "2 2 2024-08-09 18:15:23 TimeDriver oommfc None 2.00e-08 \n", "\n", " n \n", "0 NaN \n", "1 NaN \n", "2 5.00e+02 " ] }, "execution_count": 12, "metadata": {}, "output_type": "execute_result" } ], "source": [ "import micromagneticdata as mdata\n", "\n", "data = mdata.Data(system.name)\n", "data.info" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "To interactively inspect the time dependent magnetisation, we use `data[-1]` to refer to the last drive." ] }, { "cell_type": "code", "execution_count": 13, "metadata": { "tags": [ "nbval-ignore-output" ] }, "outputs": [ { "data": { "application/javascript": "(function(root) {\n function now() {\n return new Date();\n }\n\n var force = true;\n var py_version = '3.4.2'.replace('rc', '-rc.').replace('.dev', '-dev.');\n var reloading = false;\n 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handle_update_output);\n events.on('clear_output.CodeCell', handle_clear_output);\n events.on('delete.Cell', handle_clear_output);\n events.on('kernel_ready.Kernel', handle_kernel_cleanup);\n\n OutputArea.prototype.register_mime_type(EXEC_MIME_TYPE, append_mime, {\n safe: true,\n index: 0\n });\n}\n\nif (window.Jupyter !== undefined) {\n try {\n var events = require('base/js/events');\n var OutputArea = require('notebook/js/outputarea').OutputArea;\n if (OutputArea.prototype.mime_types().indexOf(EXEC_MIME_TYPE) == -1) {\n register_renderer(events, OutputArea);\n }\n } catch(err) {\n }\n}\n", "application/vnd.holoviews_load.v0+json": "" }, "metadata": {}, "output_type": "display_data" }, { "data": { "text/html": [ "" ] }, "metadata": {}, "output_type": "display_data" }, { "data": { "application/vnd.holoviews_exec.v0+json": "", "text/html": [ "
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\n", "" ] }, "metadata": { "application/vnd.holoviews_exec.v0+json": { "id": "p1002" } }, "output_type": "display_data" }, { "data": { "application/vnd.jupyter.widget-view+json": { "model_id": "2e681e004b2c48cc8c0f1b10a09a38bd", "version_major": 2, "version_minor": 0 }, "text/plain": [ "BokehModel(combine_events=True, render_bundle={'docs_json': {'97dc65d0-0af3-489f-ae28-f2361a528c61': {'version…" ] }, "execution_count": 13, "metadata": {}, "output_type": "execute_result" } ], "source": [ "data[-1].hv(\n", " kdims=[\"x\", \"y\"], vdims=[\"x\", \"y\"], scalar_kw={\"cmap\": \"viridis\", \"clim\": (0, Ms)}\n", ")" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "We can now compute winding number using operators from `discretisedfield`:\n", "$$ S = \\frac{1}{4\\pi}\\iint q \\,\\,\\text{d}x\\text{d}y = \\frac{1}{4\\pi}\\iint\\mathbf{m}\\cdot\\left(\\frac{\\partial \\mathbf{m}}{\\partial x} \\times \\frac{\\partial \\mathbf{m}}{\\partial y~}\\right)\\text{d}x\\text{d}y$$" ] }, { "cell_type": "code", "execution_count": 14, "metadata": { "tags": [] }, "outputs": [ { "data": { "text/plain": [ "array([0.42290507])" ] }, "execution_count": 14, "metadata": {}, "output_type": "execute_result" } ], "source": [ "import math\n", "\n", "m = system.m.orientation.sel(\"z\")\n", "S = m.dot(m.diff(\"x\").cross(m.diff(\"y\"))).integrate() / (4 * math.pi)\n", "S" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "The winding number is commonly used and there is a predefined function in `discretisedfield.tools`. To get more accurate results we use a different numerical method than just \"naively\" evaluating the integral." ] }, { "cell_type": "code", "execution_count": 15, "metadata": { "editable": true, "slideshow": { "slide_type": "" }, "tags": [ "nbval-ignore-output" ] }, "outputs": [ { "data": { "text/plain": [ "0.5024556893362209" ] }, "execution_count": 15, "metadata": {}, "output_type": "execute_result" } ], "source": [ "df.tools.topological_charge(system.m.sel(\"z\"), method=\"berg-luescher\")" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "We can also plot the topological charge density in an interactive plot." ] }, { "cell_type": "code", "execution_count": 16, "metadata": { "editable": true, "slideshow": { "slide_type": "" }, "tags": [ "nbval-ignore-output" ] }, "outputs": [ { "data": { "application/vnd.jupyter.widget-view+json": { "model_id": "380bc0b964a24d99b1d3755c964c0121", "version_major": 2, "version_minor": 0 }, "text/plain": [ "BokehModel(combine_events=True, render_bundle={'docs_json': {'24c4550b-78b2-4e13-a6d6-98d6d0ac7f4b': {'version…" ] }, "execution_count": 16, "metadata": {}, "output_type": "execute_result" } ], "source": [ "data[-1].register_callback(\n", " lambda f: df.tools.topological_charge_density(f.sel(\"z\"))\n", ").hv(kdims=[\"x\", \"y\"])" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### Trajectory of the vortex core\n", "\n", "We can compute the trajectory of the vortex core via the center of mass of the topological charge:\n", "$$\\mathbf{R} = \\frac{ \\int \\mathbf{r} \\rho(\\mathbf{r}) d^2\\mathbf{r}}{\\int \\rho(\\mathbf{r}) d^2\\mathbf{r}}. $$" ] }, { "cell_type": "code", "execution_count": 17, "metadata": { "editable": true, "slideshow": { "slide_type": "" }, "tags": [ "nbval-ignore-output" ] }, "outputs": [ { "data": { "text/plain": [ "array([6.24115029e-12, 4.54330907e-12])" ] }, "execution_count": 17, "metadata": {}, "output_type": "execute_result" } ], "source": [ "rho = df.tools.topological_charge_density(system.m.sel(\"z\"))\n", "r = system.m.sel(\"z\").mesh.coordinate_field()\n", "R = (r * rho).integrate() / rho.integrate()\n", "R" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Now, we need to find the center of the vortex at each time step, this can be achieved by taking the data from last drive." ] }, { "cell_type": "code", "execution_count": 18, "metadata": {}, "outputs": [], "source": [ "def compute_vortex_centre(drive):\n", " x_coords = []\n", " y_coords = []\n", "\n", " r = drive[0].sel(\"z\").mesh.coordinate_field()\n", "\n", " for m in drive:\n", " tcd = df.tools.topological_charge_density(m.sel(\"z\"))\n", " centre_of_mass = (r * tcd).integrate() / tcd.integrate()\n", " x_coords.append(centre_of_mass[0])\n", " y_coords.append(centre_of_mass[1])\n", "\n", " return pd.DataFrame(\n", " {\"t\": drive.table.data[\"t\"], \"pos x\": x_coords, \"pos y\": y_coords}\n", " )" ] }, { "cell_type": "code", "execution_count": 19, "metadata": {}, "outputs": [], "source": [ "pos_pol_plus = compute_vortex_centre(data[-1])" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "We can now plot the vortex trajectory on top of the initial configuration." ] }, { "cell_type": "code", "execution_count": 20, "metadata": { "editable": true, "slideshow": { "slide_type": "" }, "tags": [ "nbval-ignore-output" ] }, "outputs": [ { "data": { "text/plain": [ "[]" ] }, "execution_count": 20, "metadata": {}, "output_type": "execute_result" }, { "data": { "image/svg+xml": [ "\n", "\n", "\n", " \n", " \n", " \n", " \n", " 2024-08-09T18:16:05.615103\n", " image/svg+xml\n", " \n", " \n", " Matplotlib v3.9.1, https://matplotlib.org/\n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " 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" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "fig, ax = plt.subplots()\n", "data[-1][0].orientation.sel(\"z\").mpl(ax=ax, scalar_kw={\"clim\": (0, 1)})\n", "ax.plot(pos_pol_plus[\"pos x\"] * 1e9, pos_pol_plus[\"pos y\"] * 1e9, c=\"yellow\")" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Finally, let us delete all simulation files:" ] }, { "cell_type": "code", "execution_count": 21, "metadata": { "tags": [] }, "outputs": [], "source": [ "oc.delete(system)" ] } ], "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": "3.11.9" }, "widgets": { "application/vnd.jupyter.widget-state+json": { "state": {}, "version_major": 2, "version_minor": 0 } } }, "nbformat": 4, "nbformat_minor": 4 }