docs/notebooks/formulary/coulomb.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "id": "e59bc6b1",
   "metadata": {},
   "source": [
    "# Coulomb logarithms"
   ]
  },
  {
   "attachments": {
    "coulomb-collision.png": {
     "image/png": 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"
    }
   },
   "cell_type": "markdown",
   "id": "304b7591",
   "metadata": {},
   "source": [
    "[impact parameter]: https://en.wikipedia.org/wiki/Impact_parameter\n",
    "\n",
    "[Coulomb collisions](https://en.wikipedia.org/wiki/Coulomb_collision) are collisions between two charged particles where the interaction is governed solely by the electric fields from the two particles. Coulomb collisions usually result in small deflections of particle trajectories. The deflection angle depends on the [impact parameter] of the collision, which is the perpendicular distance between the particle's trajectory and the particle it is colliding with.  High impact parameter collisions (which result in small deflection angles) occur much more frequently than low impact parameter collisions (which result in large deflection angles).\n",
    "\n",
    "![coulomb-collision.png](attachment:coulomb-collision.png)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "1039816a",
   "metadata": {},
   "source": [
    "[Debye length]: https://en.wikipedia.org/wiki/Debye_length\n",
    "[impact_parameter_perp]: https://docs.plasmapy.org/en/stable/api/plasmapy.formulary.collisions.impact_parameter_perp.html#plasmapy.formulary.collisions.impact_parameter_perp\n",
    "[Debye_length]: https://docs.plasmapy.org/en/stable/api/plasmapy.formulary.lengths.Debye_length.html#plasmapy.formulary.lengths.Debye_length\n",
    "[impact_parameter]: https://docs.plasmapy.org/en/stable/api/plasmapy.formulary.collisions.impact_parameter.html#plasmapy.formulary.collisions.impact_parameter\n",
    "[plasmapy.formulary]: https://docs.plasmapy.org/en/stable/formulary/index.html\n",
    "[reduced mass]: https://en.wikipedia.org/wiki/Reduced_mass\n",
    "\n",
    "The minimum and maximum impact parameters ($b_\\min$ and $b_\\max$, respectively) represent the range of distances of closest approach. While a typical Coulomb collision results in only a slight change in trajectory, the effects of these collisions are cumulative, and it is necessary to integrate over the range of impact parameters to account for the effects of Coulomb collisions throughout the plasma. The Coulomb logarithm accounts for the range in impact parameters for the different collisions, and is given by \n",
    "$$\\ln{Λ} ≡ \\ln\\left(\\frac{b_\\max}{b_\\min}\\right).$$\n",
    "\n",
    "But what should we use for the impact parameters?\n",
    "\n",
    "Usually $b_\\max$ is given by the [Debye length], $λ_D$, which can be calculated with [Debye_length]. On length scales $≳ λ_D$, electric fields from individual particles get cancelled out due to screening effects. Consequently, Coulomb collisions with impact parameters $≳ λ_D$ will rarely occur. \n",
    "\n",
    "The inner impact parameter $b_\\min$ requires more nuance. One possibility would be to set $b_\\min$ to be the impact parameter corresponding to a 90° deflection angle, $ρ_⟂$, which can be calculated with [impact_parameter_perp].  Alternatively, $b_\\min$ could be set to be the [de Broglie wavelength](https://en.wikipedia.org/wiki/Matter_wave), $λ_{dB}$, calculated using the [reduced mass], $μ$, of the particles. Typically, $$b_\\min = \\max\\left\\{ρ_⟂, λ_{dB} \\right\\}.$$\n",
    "\n",
    "The [impact_parameter] function in [plasmapy.formulary] simultaneously calculates both $b_\\min$ and $b_\\max$. Let's estimate $b_\\min$ and $b_\\max$ for proton-electron collisions in the solar corona."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "5c4e8501",
   "metadata": {},
   "outputs": [],
   "source": [
    "%matplotlib inline\n",
    "%config InlineBackend.figure_formats = ['svg']\n",
    "\n",
    "import astropy.units as u\n",
    "import matplotlib.pyplot as plt\n",
    "import numpy as np\n",
    "\n",
    "from astropy.visualization import quantity_support\n",
    "\n",
    "from plasmapy.formulary import (\n",
    "    Coulomb_logarithm,\n",
    "    impact_parameter,\n",
    "    impact_parameter_perp,\n",
    "    thermal_speed,\n",
    ")\n",
    "from plasmapy.particles import reduced_mass"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "b69fead9",
   "metadata": {},
   "outputs": [],
   "source": [
    "solar_corona = {\n",
    "    \"T\": 1e6 * u.K,\n",
    "    \"n_e\": 1e15 * u.m**-3,\n",
    "    \"species\": [\"e-\", \"p+\"],\n",
    "}\n",
    "\n",
    "bmin, bmax = impact_parameter(**solar_corona)\n",
    "\n",
    "print(f\"{bmin = :.2e}\")\n",
    "print(f\"{bmax = :.2e}\")"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "53ba7ecf",
   "metadata": {},
   "source": [
    "When we can calculate the Coulomb logarithm, we find that it is ∼20 (a common value for astrophysical plasma)."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "7077f58d",
   "metadata": {},
   "outputs": [],
   "source": [
    "Coulomb_logarithm(**solar_corona)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "036e5a64",
   "metadata": {},
   "source": [
    "[Quantity]: https://docs.astropy.org/en/stable/api/astropy.units.Quantity.html#astropy.units.Quantity\n",
    "[quantity_support]: https://docs.astropy.org/en/stable/api/astropy.visualization.quantity_support.html\n",
    "\n",
    "Our next goals are visualize the Coulomb logarithm and impact parameters for different physical conditions.  Let's start by creating a function to plot $\\ln{Λ}$ against temperature.  We will use [quantity_support] to enable plotting with [Quantity] objects."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "f3863ccf",
   "metadata": {},
   "outputs": [],
   "source": [
    "def plot_coulomb(T, n_e, species, **kwargs):\n",
    "    \"\"\"Plot the Coulomb logarithm and impact parameter length scales.\"\"\"\n",
    "\n",
    "    ln_Λ = Coulomb_logarithm(T, n_e, species, **kwargs)\n",
    "\n",
    "    with quantity_support():\n",
    "\n",
    "        fig, ax = plt.subplots()\n",
    "\n",
    "        fig.set_size_inches(4, 4)\n",
    "\n",
    "        ax.semilogx(T, ln_Λ)\n",
    "        ax.set_xlabel(\"T (K)\")\n",
    "        ax.set_ylabel(\"$\\ln{Λ}$\")\n",
    "        ax.set_title(\"Coulomb logarithm\")\n",
    "\n",
    "        fig.tight_layout()"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "15324ea3",
   "metadata": {},
   "source": [
    "Next we will create a function to plot $b_\\min$ and $b_\\max$ as functions of temperature, along with the $ρ_⟂$ and $λ_{dB}$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "3a2b3d5d",
   "metadata": {},
   "outputs": [],
   "source": [
    "from astropy.constants import hbar as ℏ\n",
    "\n",
    "\n",
    "def plot_impact_parameters(T, n_e, species, **kwargs):\n",
    "    \"\"\"Plot the minimum & maximum impact parameters for Coulomb collisions.\"\"\"\n",
    "\n",
    "    bmin, bmax = impact_parameter(T, n_e, species, **kwargs)\n",
    "\n",
    "    μ = reduced_mass(*species)\n",
    "    V_the = thermal_speed(T, \"e-\")\n",
    "    λ_dB = ℏ / (2 * μ * V_the)\n",
    "\n",
    "    ρ_perp = impact_parameter_perp(T, species)\n",
    "\n",
    "    with quantity_support():\n",
    "        fig, ax = plt.subplots()\n",
    "        fig.set_size_inches(4, 4)\n",
    "\n",
    "        ax.loglog(T, λ_dB, \"--\", label=\"$λ_\\mathrm{dB}$\")\n",
    "        ax.loglog(T, ρ_perp, \"-.\", label=\"$ρ_\\perp$\")\n",
    "        ax.loglog(T, bmin, \"-\", label=\"$b_\\mathrm{min}$\")\n",
    "        ax.loglog(T, bmax, \"-\", label=\"$b_\\mathrm{max}$\")\n",
    "\n",
    "        ax.set_xlabel(\"T (K)\")\n",
    "        ax.set_ylabel(\"b (m)\")\n",
    "        ax.set_title(\"Impact parameters\")\n",
    "        ax.legend()\n",
    "\n",
    "        fig.tight_layout()"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "f0740c74",
   "metadata": {},
   "source": [
    "Let's now plot the Coulomb logarithm and minimum/maximum impact parameters over a range of temperatures in a dense plasma."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "a5823db1",
   "metadata": {},
   "outputs": [],
   "source": [
    "n_e = 1e25 * u.m**-3\n",
    "\n",
    "dense_plasma = {\n",
    "    \"T\": np.geomspace(1e5, 2e6) * u.K,\n",
    "    \"n_e\": n_e,\n",
    "    \"species\": [\"e-\", \"p+\"],\n",
    "}\n",
    "\n",
    "plot_coulomb(**dense_plasma)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "f19a17a7",
   "metadata": {},
   "source": [
    "To investigate what is happening at the sudden change in slope, let's plot the impact parameters against temperature."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "5611a19e",
   "metadata": {},
   "outputs": [],
   "source": [
    "plot_impact_parameters(**dense_plasma)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "d8a7ab7e",
   "metadata": {},
   "source": [
    "At low temperatures, $b_\\min$ is $ρ_⟂$. At higher temperatures, $b_\\min$ is $λ_{dB}$. What happens if we look at lower temperatures?"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "6de70259",
   "metadata": {},
   "outputs": [],
   "source": [
    "cool_dense_plasma = {\n",
    "    \"T\": np.geomspace(1e2, 2e4) * u.K,\n",
    "    \"n_e\": n_e,\n",
    "    \"species\": [\"e-\", \"p+\"],\n",
    "}\n",
    "\n",
    "plot_coulomb(**cool_dense_plasma)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "829d8dba",
   "metadata": {},
   "source": [
    "The Coulomb logarithm becomes negative! 🙀 Let's look at the impact parameters again to understand what's happening."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "241e0c66",
   "metadata": {},
   "outputs": [],
   "source": [
    "plot_impact_parameters(**cool_dense_plasma)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "2e1c17b0",
   "metadata": {},
   "source": [
    " This unphysical situation occurs because $b_\\min > b_\\max$ at low temperatures."
   ]
  },
  {
   "cell_type": "markdown",
   "id": "13af3b4c",
   "metadata": {},
   "source": [
    "[Coulomb_logarithm]: https://docs.plasmapy.org/en/stable/api/plasmapy.formulary.collisions.Coulomb_logarithm.html#coulomb-logarithm\n",
    "[Gericke, Murillo, and Schlanges (2002)]: https://doi.org/10.1103/PhysRevE.65.036418\n",
    "\n",
    "So how should we handle this?  Fortunately, PlasmaPy's implementation of [Coulomb_logarithm] includes the methods described by [Gericke, Murillo, and Schlanges (2002)] for dense, strongly-coupled plasmas.  For most cases, we recommend using `method=\"hls_full_interp\"` in which $b_\\min$ is interpolated between $λ_{dB}$ and $ρ_⟂$, and $b_\\max$ is interpolated between $λ_D$ and the ion sphere radius, $a_i ≡ \\left(\\frac{3}{4} π n_i\\right)^{⅓}$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "72802744",
   "metadata": {
    "tags": [
     "nbsphinx-thumbnail"
    ]
   },
   "outputs": [],
   "source": [
    "cool_dense_plasma[\"z_mean\"] = 1\n",
    "cool_dense_plasma[\"method\"] = \"hls_full_interp\"\n",
    "\n",
    "plot_coulomb(**cool_dense_plasma)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "87612632",
   "metadata": {},
   "source": [
    "The Coulomb logarithm approaches zero as the temperature decreases, and does not become unphysically negative. This is the expected behavior, although there is increased uncertainty for Coulomb logarithms $≲4$."
   ]
  }
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