docs/notebooks/formulary/cold_plasma_tensor_elements.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {
    "pycharm": {
     "name": "#%% md\n"
    }
   },
   "source": [
    "# Cold Magnetized Plasma Dielectric Permittivity Tensor"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "pycharm": {
     "name": "#%% raw\n"
    }
   },
   "source": [
    "This notebook shows how to calculate the values of the cold plasma tensor elements for various electromagnetic wave frequencies."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "pycharm": {
     "name": "#%%\n"
    }
   },
   "outputs": [],
   "source": [
    "%matplotlib inline"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "pycharm": {
     "name": "#%%\n"
    }
   },
   "outputs": [],
   "source": [
    "import matplotlib.pyplot as plt\n",
    "import numpy as np\n",
    "\n",
    "from astropy import units as u\n",
    "from astropy.visualization import quantity_support\n",
    "\n",
    "from plasmapy.formulary import (\n",
    "    cold_plasma_permittivity_LRP,\n",
    "    cold_plasma_permittivity_SDP,\n",
    ")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "pycharm": {
     "name": "#%% raw\n"
    }
   },
   "source": [
    "[astropy.units]: https://docs.astropy.org/en/stable/units/index.html\n",
    "[cold_plasma_permittivity_SDP]: https://docs.plasmapy.org/en/stable/api/plasmapy.formulary.dielectric.cold_plasma_permittivity_SDP.html\n",
    "[cold_plasma_permittivity_LRP]: https://docs.plasmapy.org/en/stable/api/plasmapy.formulary.dielectric.cold_plasma_permittivity_LRP.html\n",
    "\n",
    "For more information, check out the documentation on [cold_plasma_permittivity_SDP] and [cold_plasma_permittivity_LRP].\n",
    "\n",
    "Let's use [astropy.units] to define some parameters, such as the magnetic field magnitude, plasma species, and densities."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "pycharm": {
     "name": "#%%\n"
    }
   },
   "outputs": [],
   "source": [
    "B = 2 * u.T\n",
    "species = [\"e\", \"D+\"]\n",
    "n = [1e18 * u.m**-3, 1e18 * u.m**-3]"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "pycharm": {
     "name": "#%% raw\n"
    }
   },
   "source": [
    "Let's pick some frequencies in the radio frequency range."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "pycharm": {
     "name": "#%%\n"
    }
   },
   "outputs": [],
   "source": [
    "f_min = 1 * u.MHz\n",
    "f_max = 200 * u.GHz\n",
    "\n",
    "f = np.geomspace(f_min, f_max, num=5000).to(u.Hz)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "pycharm": {
     "name": "#%% raw\n"
    }
   },
   "source": [
    "[equivalency]: https://docs.astropy.org/en/stable/units/equivalencies.html\n",
    "[angular frequencies]: https://docs.plasmapy.org/en/stable/development/code_guide.html#angular-frequencies\n",
    "\n",
    "Next we convert these frequencies to angular frequencies. To do this we must specify an [equivalency] between a cycle per second and a hertz. The reasons why are described in the section of PlasmaPy's documentation on [angular frequencies]."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "pycharm": {
     "name": "#%%\n"
    }
   },
   "outputs": [],
   "source": [
    "ω_RF = f.to(u.rad / u.s, equivalencies=[(u.Hz, u.cycle / u.s)])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "pycharm": {
     "name": "#%% raw\n"
    }
   },
   "source": [
    "[Stix 1992]: https://link.springer.com/book/9780883188590\n",
    "\n",
    "In a Jupyter notebook, we can type `\\omega` and press tab to get \"ω\".\n",
    "\n",
    "Now we are ready to calculate the $S$ (sum), $D$ (difference), and $P$ (plasma) components of the dielectric tensor in the \"Stix\" frame with $B ∥ ẑ$.  This notation is from [Stix 1992]."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "pycharm": {
     "name": "#%%\n"
    }
   },
   "outputs": [],
   "source": [
    "S, D, P = cold_plasma_permittivity_SDP(B=B, species=species, n=n, omega=ω_RF)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "pycharm": {
     "name": "#%% raw\n"
    }
   },
   "source": [
    "Next we will filter the negative and positive values so that they can be plotted separately. We'll be doing this multiple times, so let's create a function using the `numpy.ma` module for working with masked arrays."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "pycharm": {
     "name": "#%%\n"
    }
   },
   "outputs": [],
   "source": [
    "def filter_negative_and_positive_values(arr):\n",
    "    \"\"\"\n",
    "    Return an array with only negative values of ``arr``\n",
    "    and an array with only positive values of ``arr``.\n",
    "    Each element of these arrays that does not meet the\n",
    "    condition are replaced with `~numpy.nan`.\n",
    "    \"\"\"\n",
    "    arr_neg = np.ma.masked_greater_equal(arr, 0).filled(np.nan)\n",
    "    arr_pos = np.ma.masked_less_equal(arr, 0).filled(np.nan)\n",
    "    return arr_neg, arr_pos"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "pycharm": {
     "name": "#%% raw\n"
    }
   },
   "source": [
    "Let's apply this function to `S`, `D`, and `P`."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "pycharm": {
     "name": "#%%\n"
    }
   },
   "outputs": [],
   "source": [
    "S_neg, S_pos = filter_negative_and_positive_values(S)\n",
    "D_neg, D_pos = filter_negative_and_positive_values(D)\n",
    "P_neg, P_pos = filter_negative_and_positive_values(P)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "pycharm": {
     "name": "#%% raw\n"
    }
   },
   "source": [
    "[Quantity]: https://docs.astropy.org/en/stable/units/quantity.html\n",
    "[quantity_support]: https://docs.astropy.org/en/stable/api/astropy.visualization.quantity_support.html?highlight=quantity_support#quantity-support\n",
    "\n",
    "Because we are using [Quantity] objects, we will need to call [quantity_support] before plotting the results."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "pycharm": {
     "name": "#%%\n"
    }
   },
   "outputs": [],
   "source": [
    "quantity_support()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "pycharm": {
     "name": "#%% md\n"
    }
   },
   "source": [
    "While we're at it, let's specify a few colorblind friendly colors to use in the plots."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "pycharm": {
     "name": "#%%\n"
    }
   },
   "outputs": [],
   "source": [
    "red, blue, orange = \"#920000\", \"#006ddb\", \"#db6d00\""
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "pycharm": {
     "name": "#%% raw\n"
    }
   },
   "source": [
    "Let's plot the elements of the cold plasma dielectric tensor in the Stix frame. "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "pycharm": {
     "name": "#%%\n"
    }
   },
   "outputs": [],
   "source": [
    "ylim = (1e-4, 1e8)\n",
    "\n",
    "plt.figure(figsize=(12, 6))\n",
    "\n",
    "plt.semilogx(f, abs(S_neg), red, lw=2, ls=\"--\", label=\"S < 0\")\n",
    "plt.semilogx(f, abs(D_neg), blue, lw=2, ls=\"--\", label=\"D < 0\")\n",
    "plt.semilogx(f, abs(P_neg), orange, lw=2, ls=\"--\", label=\"P < 0\")\n",
    "\n",
    "plt.semilogx(f, S_pos, red, lw=2, label=\"S > 0\")\n",
    "plt.semilogx(f, D_pos, blue, lw=2, label=\"D > 0\")\n",
    "plt.semilogx(f, P_pos, orange, lw=2, label=\"P > 0\")\n",
    "\n",
    "plt.ylim(ylim)\n",
    "plt.xlim(f_min, f_max)\n",
    "\n",
    "plt.title(\n",
    "    \"Cold plasma dielectric permittivity tensor components in Stix frame\", size=16\n",
    ")\n",
    "\n",
    "plt.xlabel(\"Frequency [Hz]\", size=16)\n",
    "plt.ylabel(\"Absolute value\", size=16)\n",
    "\n",
    "plt.yscale(\"log\")\n",
    "\n",
    "plt.grid(True, which=\"major\")\n",
    "plt.grid(True, which=\"minor\")\n",
    "\n",
    "plt.tick_params(labelsize=14)\n",
    "\n",
    "plt.legend(fontsize=18, ncol=2, framealpha=1)\n",
    "\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "pycharm": {
     "name": "#%% raw\n"
    }
   },
   "source": [
    "[cold_plasma_permittivity_LRP]: https://docs.plasmapy.org/en/stable/api/plasmapy.formulary.dielectric.cold_plasma_permittivity_LRP.html\n",
    "\n",
    "Next let's get the dielectric permittivity tensor elements in the \"rotating\" basis where the tensor is diagonal and with $B ∥ ẑ$. We will use [cold_plasma_permittivity_LRP] to get the left-handed circular polarization tensor element $L$, the right-handed circular polarization tensor element $R$, and the plasma component $P$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "pycharm": {
     "name": "#%%\n"
    }
   },
   "outputs": [],
   "source": [
    "L, R, P = cold_plasma_permittivity_LRP(B, species, n, ω_RF)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "pycharm": {
     "name": "#%% raw\n"
    }
   },
   "source": [
    "Let's use the function from earlier to prepare for plotting."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "pycharm": {
     "name": "#%%\n"
    }
   },
   "outputs": [],
   "source": [
    "L_neg, L_pos = filter_negative_and_positive_values(L)\n",
    "R_neg, R_pos = filter_negative_and_positive_values(R)\n",
    "P_neg, P_pos = filter_negative_and_positive_values(P)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "pycharm": {
     "name": "#%% raw\n"
    }
   },
   "source": [
    "Now we can plot the tensor components rotating frame too."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "pycharm": {
     "name": "#%%\n"
    }
   },
   "outputs": [],
   "source": [
    "plt.figure(figsize=(12, 6))\n",
    "\n",
    "plt.semilogx(f, abs(L_neg), red, lw=2, ls=\"--\", label=\"L < 0\")\n",
    "plt.semilogx(f, abs(R_neg), blue, lw=2, ls=\"--\", label=\"R < 0\")\n",
    "plt.semilogx(f, abs(P_neg), orange, lw=2, ls=\"--\", label=\"P < 0\")\n",
    "\n",
    "plt.semilogx(f, L_pos, red, lw=2, label=\"L > 0\")\n",
    "plt.semilogx(f, R_pos, blue, lw=2, label=\"R > 0\")\n",
    "plt.semilogx(f, P_pos, orange, lw=2, label=\"P > 0\")\n",
    "\n",
    "plt.ylim(ylim)\n",
    "plt.xlim(f_min, f_max)\n",
    "\n",
    "plt.title(\n",
    "    \"Cold plasma dielectric permittivity tensor components in the rotating frame\",\n",
    "    size=16,\n",
    ")\n",
    "\n",
    "plt.xlabel(\"Frequency [Hz]\", size=16)\n",
    "plt.ylabel(\"Absolute value\", size=16)\n",
    "\n",
    "plt.yscale(\"log\")\n",
    "\n",
    "plt.grid(True, which=\"major\")\n",
    "plt.grid(True, which=\"minor\")\n",
    "\n",
    "plt.tick_params(labelsize=14)\n",
    "\n",
    "plt.legend(fontsize=18, ncol=2, framealpha=1)\n",
    "\n",
    "plt.show()"
   ]
  }
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