{
 "cells": [
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "import numpy as np\n",
    "import matplotlib.pyplot as plt"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "τ = 1\n",
    "N0 = 1\n",
    "t = np.linspace(0,6*τ)\n",
    "\n",
    "plt.rc('font', size=14)\n",
    "plt.plot(t/τ, N0*np.exp(-t/τ), label='')\n",
    "plt.xlabel('t/τ')\n",
    "plt.ylabel(r'N/N$_0$')\n",
    "plt.ylim([0,1])\n",
    "plt.xlim([0,4])\n",
    "plt.plot([0,1],[1,0], '-', color='orange', lw=1, label='τ');\n",
    "plt.plot([0,τ*np.log(2)],[0.5,0.5], '-', color='green', lw=1, label=r't$_{1/2}$');\n",
    "plt.plot([τ*np.log(2),τ*np.log(2)],[0.5,0], '-', color='green', lw=1);\n",
    "plt.legend(frameon=False);\n",
    "plt.savefig('halflife.svg')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  }
 ],
 "metadata": {
  "kernelspec": {
   "display_name": "Python 3",
   "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": "3.8.5"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 4
}
