RutherfordScatteringZPIC.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Rutherford scattering\n",
    "\n",
    "Notebook by: Óscar Amaro (2020)\n",
    "\n",
    "In this notebook we show how to obtain the Rutherford cross section in ZPIC.\n",
    "\n",
    "ZPIC does not have a collision module, but we can simulate collisions with an external Coulomb field. This means this notebook does not demonstrate any plasma physics, but rather a purely classical collision."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Links:\n",
    "- Check out [ZPIC on Github](https://github.com/zpic-plasma/zpic)\n",
    "- And your account on [ZPIC server](https://zpic.tecnico.ulisboa.pt/hub/login)\n",
    "\n",
    "Other links:\n",
    "- Rutherford scattering on [Wikipedia](https://en.wikipedia.org/wiki/Rutherford_scattering)\n",
    "\n",
    "Bibliography:\n",
    "- Callen [Chapter 2](http://homepages.cae.wisc.edu/~callen/book.html)\n",
    "- Goldstein 'Classical Mechanics' §3.10"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "In this notebook you will learn:\n",
    "- scattering from coulomb field\n",
    "- variable normalization\n",
    "- apply external field\n",
    "- define custom density profiles\n",
    "- turn off current deposition (downgrading zpic to a particle pusher)\n",
    "- build histograms and compare with theoretical cross-section"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "To consolidate what you've learned about rutherford scattering, the notebook includes some exercises."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "from __future__ import print_function\n",
    "from ipywidgets import interact, interactive, fixed, interact_manual\n",
    "import ipywidgets as widgets\n",
    "from IPython.display import display\n",
    "from IPython.display import HTML\n",
    "\n",
    "import os,sys,inspect\n",
    "currentdir = os.path.dirname(os.path.abspath(inspect.getfile(inspect.currentframe())))\n",
    "parentdir = os.path.dirname(currentdir)\n",
    "sys.path.insert(0,parentdir) \n",
    "import em2d\n",
    "import numpy as np\n",
    "import matplotlib.pyplot as plt\n",
    "import matplotlib.colors as colors\n",
    "from matplotlib import animation, rc\n",
    "import math\n",
    "# style\n",
    "fntsz=20"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Notation\n",
    "\n",
    "- $b$ is the impact parameter\n",
    "- $\\theta$ is the scattering angle \n",
    "- $d\\sigma/d\\Omega$  is the differential cross-section\n",
    "- $\\phi$ is the azimuthal angle\n",
    "\n",
    "The schematic below was taken from Griffiths' 'Quantum Mechanics', Chapter 11."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "![title](scattering.png)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### __Normalization__\n",
    "Before beginning it is important to normalize the relevant equations, starting with the Coulomb field.\n",
    "$$E=\\frac{Z e}{r^2} \\Leftrightarrow \\frac{m_e c \\omega_p}{e} E' = \\frac{\\omega_p^2 e}{c^2} \\frac{Z}{r'^2} \\Leftrightarrow E' = \\frac{\\omega_p e^2}{m_e c^3} \\frac{Z}{r'^2}$$\n",
    "\n",
    "now, as $\\omega_p^2 = n_e e^2/m_e$ and $[n_e] = L^{-3} = \\omega_p^3 c^{-3}$ we have $\\omega_p^2 = \\frac{\\omega_p^3 c^{-3} e^2}{m_e} \\Rightarrow \\omega_p = \\frac{m_e c^3}{e^2}$\n",
    "\n",
    "Therefore $$ E' = \\frac{Z}{r'^2}$$\n",
    "\n",
    "### __Impact parameter__\n",
    "In the case of Rutherford scattering we have\n",
    "$$\n",
    "b=\\left(q_{1} q_{2} / 8 \\pi \\epsilon_{0} \\epsilon \\right) \\cot (\\theta / 2)\n",
    "$$\n",
    "where $\\epsilon=\\frac{1}{2}m_e v_0^2$ is the kinetic energy and $v_0$ is the velocity of the incoming particle. We can rewrite this as\n",
    "$$b'= \\frac{z Z}{v_0'^2} \\frac{1}{\\tan \\theta/2}$$\n",
    "\n",
    "### __Cross section__\n",
    "The differential cross-section is defined as\n",
    "$$\n",
    "\\frac{d\\sigma}{d\\Omega}=\\frac{b}{\\sin \\theta}\\left|\\frac{d b}{d \\theta}\\right|\n",
    "$$\n",
    "\n",
    "In the case of Rutherford scattering we have\n",
    "$$\n",
    "\\frac{d\\sigma}{d\\Omega}=\\left[\\frac{q_{1} q_{2}}{16 \\pi \\epsilon_{0} \\epsilon \\sin ^{2}(\\theta / 2)}\\right]^{2}\n",
    "$$\n",
    "\n",
    "but instead of normalizing the equation, the particle histogram will be normalized numerically."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Exercise 1\n",
    "\n",
    "In order to run the simulation and draw some animations we need to turn off some lines of code that do not change the physics dramatically.\n",
    "The meaning of these changes will become clearer later on.\n",
    "\n",
    "Please follow these instructions:\n",
    "- Turn off current deposition. Go to the `em2d` folder, run `make clean`. In `simulation.c`, comment out the lines `current_update( &sim -> current );` and `emf_advance( &sim -> emf, &sim -> current );`\n",
    "- Turn off particle sorting. In `particles.c`, comment out the line `if ( ! (spec -> iter % 16) ) spec_sort( spec );` . Run `make`\n",
    "- Go to the `python` folder. Run `make clean`, `make`\n",
    "- We are ready to go!"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We start by shooting only one electron"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "application/vnd.jupyter.widget-view+json": {
       "model_id": "468844561f1f4407bdbc21920f502f4a",
       "version_major": 2,
       "version_minor": 0
      },
      "text/plain": [
       "interactive(children=(IntSlider(value=5, description='b', max=10), Output()), _dom_classes=('widget-interact',…"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "def chooseb(b):\n",
    "    #display(b)\n",
    "    return b\n",
    "pywidg = interactive(chooseb, b=widgets.IntSlider(min=0, max=10, step=1, value=5))\n",
    "display(pywidg)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "#electrons= 1\n"
     ]
    }
   ],
   "source": [
    "# Grid\n",
    "nxx = 32*6\n",
    "nyy = 128*4*4\n",
    "# box\n",
    "boxL = 12.8*4; boxH = 35\n",
    "xsource = boxL/2.0; ysource = boxH/6.0\n",
    "xbeam = boxL/20; ybeam=ysource+pywidg.result;\n",
    "beamL = boxL/nxx; beamH = boxH/nyy\n",
    "# scatterer charge\n",
    "Z = -0.3\n",
    "# Time step\n",
    "ddt = 10 / ( 1/(boxL/nxx)**2 + 1/(boxH/nyy)**2 )\n",
    "tmax = 65.0\n",
    "titer = np.int(tmax/ddt)\n",
    "# initial beam momentum\n",
    "ufl0 = 0.5;\n",
    "\n",
    "# Custom density profiles along x and y\n",
    "def xdensity(x):\n",
    "    return (np.heaviside(x-(xbeam-beamL),0.5)-np.heaviside(x-(xbeam),0.5))\n",
    "def ydensity(y):\n",
    "    return (np.heaviside(y-(ybeam),0.5)-np.heaviside(y-(ybeam+beamH),0.5))\n",
    "\n",
    "# Background plasma\n",
    "density = em2d.Density( type = \"custom\", custom_x = xdensity, custom_y = ydensity )\n",
    "electrons = em2d.Species( \"electrons\", -1.0, ppc = [2,2], ufl = [ufl0,0.0,0.0], density = density )\n",
    "# Initialize simulation data\n",
    "sim = em2d.Simulation( nx = [nxx,nyy], box = [boxL,boxH], dt = ddt, \n",
    "                       species = electrons )\n",
    "npart = len(electrons.particles['x'])\n",
    "print('#electrons=',npart)\n",
    "\n",
    "# create array to store the trajectories of the particles\n",
    "trajectory = np.zeros((21,npart,2))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Exercise 3\n",
    "- Write down the right form of the electric field"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [],
   "source": [
    "# Electric field\n",
    "def ext_E1(ix,dx,iy,dy):\n",
    "    x = ix*dx - xsource;\n",
    "    y = iy*dy - ysource;\n",
    "    r=math.sqrt(x**2+y**2)\n",
    "    phi=math.atan2(x,y)\n",
    "    mod_r = Z/r**2\n",
    "    return [mod_r*math.sin(phi),mod_r*math.cos(phi),0]\n",
    "# Magnetic field\n",
    "def ext_B1(ix,dx,iy,dy):\n",
    "    return [0.0,0.0,0.0]\n",
    "# apply fields to simulation\n",
    "ext = em2d.ExternalField(E_type = 'custom',E_custom=ext_E1,B_type = 'custom',B_custom=ext_B1)\n",
    "sim.emf.set_ext_fld( ext )"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "n = 22345, t = 64.9945,  65\r"
     ]
    }
   ],
   "source": [
    "# Run the simulation\n",
    "while sim.n < titer:\n",
    "    print('n = {:d}, t = {:g},  {:g}'.format(sim.n,sim.t,tmax), end = '\\r')\n",
    "    if np.mod(sim.n,np.int(titer/20))==0:\n",
    "        electronx = (electrons.particles['ix'] + electrons.particles['x']) * boxL/nxx\n",
    "        electrony = (electrons.particles['iy'] + electrons.particles['y']) * boxH/nyy\n",
    "        trajectory[np.int(sim.n/np.int(titer/20))-1,:,:] = np.transpose(np.array([electronx,electrony]))\n",
    "    sim.iter()\n",
    "# save last \n",
    "electronx = (electrons.particles['ix'] + electrons.particles['x']) * boxL/nxx\n",
    "electrony = (electrons.particles['iy'] + electrons.particles['y']) * boxH/nyy\n",
    "trajectory[-1,:,:] = np.transpose(np.array([electronx,electrony]))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "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": [
    "# plot the trajectory\n",
    "plt.plot(trajectory[:,0,0],trajectory[:,0,1],color='#1f77b4',label='electron')\n",
    "plt.scatter([xsource],[ysource],s=200,label='center')\n",
    "plt.xlim([0,boxL])\n",
    "plt.ylim([0,boxH])\n",
    "plt.xlabel(r'$x[c/\\omega_p]$',fontsize=fntsz)\n",
    "plt.ylabel(r'$y[c/\\omega_p]$',fontsize=fntsz)\n",
    "plt.title('Trajectory b='+str(pywidg.result)+'$c/\\omega_p$',fontsize=fntsz)\n",
    "plt.xticks(fontsize=fntsz)\n",
    "plt.yticks(fontsize=fntsz)\n",
    "plt.legend(fontsize=fntsz/1.5)\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 36,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<video width=\"432\" height=\"288\" controls autoplay loop>\n",
       "  <source type=\"video/mp4\" src=\"data:video/mp4;base64,AAAAIGZ0eXBNNFYgAAACAE00ViBpc29taXNvMmF2YzEAAAAIZnJlZQAAIXdtZGF0AAACoAYF//+c\n",
       "3EXpvebZSLeWLNgg2SPu73gyNjQgLSBjb3JlIDE1MiAtIEguMjY0L01QRUctNCBBVkMgY29kZWMg\n",
       "LSBDb3B5bGVmdCAyMDAzLTIwMTcgLSBodHRwOi8vd3d3LnZpZGVvbGFuLm9yZy94MjY0Lmh0bWwg\n",
       "LSBvcHRpb25zOiBjYWJhYz0xIHJlZj0zIGRlYmxvY2s9MTowOjAgYW5hbHlzZT0weDM6MHgxMTMg\n",
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       "\">\n",
       "  Your browser does not support the video tag.\n",
       "</video>"
      ],
      "text/plain": [
       "<matplotlib.animation.FuncAnimation at 0x11dacd7b8>"
      ]
     },
     "execution_count": 36,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# First set up the figure, the axis, and the plot element we want to animate\n",
    "fig, ax = plt.subplots()\n",
    "ax.set_xlim([0,boxL])\n",
    "ax.set_ylim([0,boxH])\n",
    "line, = ax.plot([], [], lw=2)\n",
    "ax.plot(trajectory[:,0,0],trajectory[:,0,1],color='#1f77b4',label='electron',alpha=0.2)\n",
    "ax.scatter([xsource],[ysource],s=200,label='center')\n",
    "ax.set_xlabel(r'$x[c/\\omega_p]$',fontsize=fntsz)\n",
    "ax.set_ylabel(r'$y[c/\\omega_p]$',fontsize=fntsz)\n",
    "ax.set_title('Trajectory b='+str(pywidg.result)+'$c/\\omega_p$',fontsize=fntsz)\n",
    "ax.tick_params(labelsize='medium')\n",
    "ax.legend(fontsize=fntsz/1.5)\n",
    "plt.close()\n",
    "# initialization function: plot the background of each frame\n",
    "def init():\n",
    "    #line.set_data([], [])\n",
    "    #return (line,)\n",
    "    return ()\n",
    "# animation function. This is called sequentially\n",
    "def animate(i):\n",
    "    line.set_data(trajectory[max(0,i-5):i,0,0], trajectory[max(0,i-5):i,0,1])\n",
    "    return (line,)\n",
    "# call the animator. blit=True means only re-draw the parts that have changed.\n",
    "anim = animation.FuncAnimation(fig, animate, init_func=init,\n",
    "                               frames=len(trajectory[:,0,0]), interval=60, blit=True);\n",
    "rc('animation', html='html5')\n",
    "anim"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Exercise 4\n",
    "- Try choosing different initial positions of the incoming particle"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We now include many particles in our simulation."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "gamma = 1.118033988749895 ,v/c= 0.4472135954999579\n",
      "dt = 0.0029086812545243163\n",
      "#iterations = 22346\n"
     ]
    }
   ],
   "source": [
    "# Grid\n",
    "nxx = 32*6\n",
    "nyy = 128*4*4\n",
    "# box\n",
    "boxL = 12.8*4\n",
    "boxH = 35\n",
    "xsource = boxL/2.0\n",
    "ysource = boxH/6.0\n",
    "xbeam = boxL/20\n",
    "beamL = boxL/nxx\n",
    "beamH = 0.6*boxH\n",
    "nbins = 40 # #bins for histogram\n",
    "\n",
    "# scatterer\n",
    "Z = -0.3\n",
    "\n",
    "# Time step\n",
    "ddt = 10 / ( 1/(boxL/nxx)**2 + 1/(boxH/nyy)**2 )\n",
    "tmax = 65.0\n",
    "titer = np.int(tmax/ddt)\n",
    "\n",
    "# momentum\n",
    "ufl0 = 0.5;\n",
    "\n",
    "print('gamma =',np.sqrt(ufl0**2+1),',v/c=',ufl0/np.sqrt(ufl0**2+1))\n",
    "print('dt =',ddt); print('#iterations =',titer)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Exercise 2\n",
    "- What is the time step and what is the number of iterations?\n",
    "> Hint: add ``` print('dt =',ddt); print('#iterations =',titer) ``` to the code block above\n",
    "- What is the Lorentz factor? Are we in the relativistic regime?\n",
    "> Hint: add ``` print('gamma =',np.sqrt(ufl0**2+1),',v/c=',ufl0/np.sqrt(ufl0**2+1)) ``` to the code block above"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Define density functions"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "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": [
    "# Custom density profiles along x and y\n",
    "def xdensity(x):\n",
    "    return (np.heaviside(x-(xbeam-beamL),0.5)-np.heaviside(x-(xbeam),0.5))\n",
    "def ydensity(y):\n",
    "    return (np.heaviside(y-(ysource),0.5)-np.heaviside(y-(ysource+beamH),0.5))*np.sqrt(abs((y-ysource)/(beamH-beamH/1000)))\n",
    "\n",
    "xlst = np.linspace(0,boxL,128)\n",
    "plt.plot(xlst, xdensity(xlst))\n",
    "\n",
    "ylst = np.linspace(0,boxH,128)\n",
    "plt.plot(ylst, ydensity(ylst))\n",
    "plt.xlabel(r'$x,y[c/\\omega_p]$',fontsize=fntsz)\n",
    "plt.ylabel(r'$n_e/n_0$',fontsize=fntsz)\n",
    "plt.title(r'density profiles',fontsize=fntsz)\n",
    "plt.xlim([0,boxL])\n",
    "plt.xticks(fontsize=fntsz)\n",
    "plt.yticks(fontsize=fntsz)\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "__Compare potential with kinetic energies__\n",
    "\n",
    "If the potential component is much smaller, we can say the particles start at \"infinity\".\n",
    "\n",
    "Normalization is trivial\n",
    "$$ U' = \\frac{Z}{r'} \\ \\text{and} \\ K' = \\frac{v'^2}{2}$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "0.012796471585345562\n",
      "0.09999999999999999\n"
     ]
    }
   ],
   "source": [
    "r = np.sqrt((ybeam-ysource)**2+(xbeam-xsource)**2)\n",
    "def ext_V1(r):\n",
    "    return np.abs(Z)/r\n",
    "# potential energy\n",
    "print(max(ext_V1(r)))\n",
    "# kinetic energy\n",
    "print( 0.5*(ufl0/np.sqrt(ufl0**2+1))**2 )"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {},
   "outputs": [],
   "source": [
    "# auxiliary function for plotting histograms\n",
    "def histline(x):\n",
    "    res=np.zeros((len(x)-1,1));\n",
    "    for i in range(0,len(x)-1):\n",
    "        res[i] = (x[i]+x[i+1])/2;\n",
    "    return res"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "#electrons= 183389\n"
     ]
    }
   ],
   "source": [
    "# Background plasma\n",
    "density = em2d.Density( type = \"custom\", custom_x = xdensity, custom_y = ydensity )\n",
    "electrons = em2d.Species( \"electrons\", -1.0, ppc = [16,16], ufl = [ufl0,0.0,0.0], density = density )\n",
    "\n",
    "# Initialize simulation data\n",
    "sim = em2d.Simulation( nx = [nxx,nyy], box = [boxL,boxH], dt = ddt, \n",
    "                       species = electrons )\n",
    "npart = len(electrons.particles['x'])\n",
    "print('#electrons=',npart) #remove"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "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": [
    "# initial linear density is square root\n",
    "# get impact parameter\n",
    "blst = (electrons.particles['iy'] + electrons.particles['y']) * boxH/nyy -ysource\n",
    "blst2 = np.linspace(min(blst),max(blst)) # array in range of blst\n",
    "\n",
    "plt.hist(blst,bins=nbins);\n",
    "plt.plot(blst2,np.sqrt(blst2/max(blst2))*npart/(2/3)/nbins)\n",
    "#plt.hist(trajectory[0,:,1],bins=50);\n",
    "plt.title('initial linear density should be square root',fontsize=fntsz)\n",
    "plt.xlabel(r'b',fontsize=fntsz)\n",
    "plt.ylabel(r'# electrons',fontsize=fntsz)\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "memory(trajectory) = 61618704 bytes= 0.061618704 GB\n"
     ]
    }
   ],
   "source": [
    "# create array to store the trajectories of the particles\n",
    "trajectory = np.zeros((21,npart,2))\n",
    "print(\"memory(trajectory) =\",trajectory.nbytes,'bytes=',trajectory.nbytes*1e-9,'GB') #remove"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "# Run the simulation\n",
    "while sim.n < titer:\n",
    "    print('n = {:d}, t = {:g},  {:g}'.format(sim.n,sim.t,tmax), end = '\\r')\n",
    "    if np.mod(sim.n,np.int(titer/20))==0:\n",
    "        electronx = (electrons.particles['ix'] + electrons.particles['x']) * boxL/nxx\n",
    "        electrony = (electrons.particles['iy'] + electrons.particles['y']) * boxH/nyy\n",
    "        trajectory[np.int(sim.n/np.int(titer/20))-1,:,:] = np.transpose(np.array([electronx,electrony]))\n",
    "    sim.iter()\n",
    "# save last \n",
    "electronx = (electrons.particles['ix'] + electrons.particles['x']) * boxL/nxx\n",
    "electrony = (electrons.particles['iy'] + electrons.particles['y']) * boxH/nyy\n",
    "trajectory[-1,:,:] = np.transpose(np.array([electronx,electrony]))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 249,
   "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": [
    "# plot some trajectories\n",
    "# if the number of particles or the number of time steps is too high, plotting is slowed down\n",
    "# select only 10 representative particles and 20 time steps\n",
    "slct = 10\n",
    "plt.plot(trajectory[:,0,0],trajectory[:,0,1],color='#1f77b4',label='electron')\n",
    "for i in range(1,slct):\n",
    "    plt.plot(trajectory[:,np.int(i*npart/10),0],trajectory[:,np.int(i*npart/10),1],color='#1f77b4')\n",
    "plt.scatter([xsource],[ysource],s=200,label='center')\n",
    "plt.xlim([0,boxL])\n",
    "plt.ylim([0,boxH])\n",
    "plt.xlabel(r'$x[c/\\omega_p]$',fontsize=fntsz)\n",
    "plt.ylabel(r'$y[c/\\omega_p]$',fontsize=fntsz)\n",
    "plt.title('representative trajectories',fontsize=fntsz)\n",
    "plt.xticks(fontsize=fntsz)\n",
    "plt.yticks(fontsize=fntsz)\n",
    "plt.legend(fontsize=fntsz/1.5)\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 262,
   "metadata": {},
   "outputs": [],
   "source": [
    "# create array for angle\n",
    "thetalst = np.zeros(npart)\n",
    "for i in range(npart):\n",
    "    #deltax = electrons.particles['ux'][i]\n",
    "    #deltay = electrons.particles['uy'][i]\n",
    "    \n",
    "    deltax = (trajectory[-1,i,0]-trajectory[-2,i,0])\n",
    "    deltay = (trajectory[-1,i,1]-trajectory[-2,i,1])\n",
    "    thetalst[i] = np.arctan2(deltay,deltax)\n",
    "\n",
    "# create array for the theoretical estimation\n",
    "thetalst2 = np.linspace(1e-3,np.pi)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 268,
   "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": [
    "# theta(b)\n",
    "plt.scatter(1.3*blst[0:-1:np.int(npart/1000)], thetalst[0:-1:np.int(npart/1000)],s=1,label='zpic')\n",
    "plt.plot((-1*Z/(ufl0/np.sqrt(ufl0**2+1))**2)/np.tan(thetalst2/2), thetalst2, ls = '-.', color = '#ff7f0e', label='theory',zorder=-11)\n",
    "plt.plot([min(0,min(blst)),max(blst)],[np.pi,np.pi],'--',color='#2ca02c',label=r'$\\pi,\\pi/2$')\n",
    "plt.plot([min(0,min(blst)),max(blst)],[np.pi/2,np.pi/2],'--',color='#2ca02c')\n",
    "plt.plot([(-1*Z/(ufl0/np.sqrt(ufl0**2+1))**2),(-1*Z/(ufl0/np.sqrt(ufl0**2+1))**2)],[0,np.pi],'--',color='red') # b90\n",
    "#plt.xscale('log')\n",
    "plt.ylim([0,np.pi*1.1])\n",
    "plt.xlabel(r'$b[c/\\omega_p]$',fontsize=fntsz)\n",
    "plt.ylabel(r'$\\theta$',fontsize=fntsz)\n",
    "plt.xlim([1e-1,max(blst)])\n",
    "handles, labels = plt.gca().get_legend_handles_labels()\n",
    "order = [2,0,1]\n",
    "plt.legend([handles[idx] for idx in order],[labels[idx] for idx in order],fontsize=fntsz/1.5)\n",
    "plt.xticks(fontsize=fntsz/1.5)\n",
    "plt.yticks(fontsize=fntsz/1.5)\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 264,
   "metadata": {},
   "outputs": [],
   "source": [
    "# histograms output the bin content and the bin edges (size of bin content+1), which means one cannot immediately plot these two together\n",
    "# to better compare simulation with theory, we take the output of the histogram and average the bin edges\n",
    "nbins=50\n",
    "thetamin=0.4\n",
    "distY, distX, _ = plt.hist(thetalst[thetalst>thetamin], bins=nbins)\n",
    "plt.close() # don't show histogram yet\n",
    "distX2 = histline(distX)\n",
    "\n",
    "# theoretical cross section/histogram\n",
    "dsdO = np.sin(thetalst2)/np.sin(thetalst2/2)**4"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 265,
   "metadata": {},
   "outputs": [],
   "source": [
    "from scipy.integrate import quad\n",
    "def integrand(thet):\n",
    "    return np.sin(thet)/np.sin(thet/2)**4\n",
    "dsdOnrm = quad(integrand,distX2[0,0] , distX2[-1,0], args=())[0]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 266,
   "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": [
    "plt.plot(distX2,distY,'.b',label='zpic') #/max(dsdO)*max(distY)\n",
    "plt.plot(thetalst2,dsdO/dsdOnrm*np.sum(distY)/nbins*np.pi,ls = '-.', color = '#ff7f0e',label='theory')\n",
    "#plt.yscale('log') # log scale shows mismatchings better\n",
    "plt.xlabel(r'$\\theta$',fontsize=fntsz)\n",
    "plt.ylabel(r'counts, $d\\sigma/d\\Omega$',fontsize=fntsz)\n",
    "plt.xticks(fontsize=fntsz/1.5)\n",
    "plt.yticks(fontsize=fntsz/1.5)\n",
    "plt.xlim([thetamin,np.pi]) # we are not interested in angles very close to zero, unless you ran a simulation with that intent\n",
    "plt.ylim([1,max(distY)]) # use this when vertical array spans several orders of magnitude\n",
    "plt.legend(fontsize=fntsz/1.5)\n",
    "plt.show()"
   ]
  },
  {
   "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": "3.7.4"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 4
}