Scattering.ipynb

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   "source": [
    "<div class=\"alert alert-block alert-info\">\n",
    "<h1>The Scattering Problem</h1>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "# nbi:hide_in\n",
    "import nbinteract as nbi\n",
    "import numpy as np\n",
    "import scipy as sp \n",
    "import matplotlib.pyplot as plt\n",
    "%matplotlib inline \n",
    "from ipywidgets import interact, interactive, fixed, interact_manual"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "\n",
    "## Scattering on hard sphere"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "\n",
    "Scattering angle:\n",
    "$$ \\theta(b) = 2\\arccos{\\left(\\frac{b}{R}\\right)} $$\n",
    "\n",
    "Turning point:\n",
    "$$r_{tp}=\\sqrt{R^2-b^2} $$\n",
    "\n",
    "Differential scattering cross section:\n",
    "$$ \\frac{d\\sigma}{d\\Omega} = \\frac{R^2}{4}  $$\n",
    "\n",
    "Total scattering cross section:\n",
    "$$ \\sigma = \\pi R^2 $$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Scattering Trajectory"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 27,
   "metadata": {},
   "outputs": [
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      "text/plain": [
       "interactive(children=(FloatSlider(value=1.5, description='b', max=3.0), FloatSlider(value=1.75, description='R…"
      ]
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     "metadata": {},
     "output_type": "display_data"
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    {
     "data": {
      "text/plain": [
       "<function __main__.scattering_on_hard_sphere(b, R)>"
      ]
     },
     "execution_count": 27,
     "metadata": {},
     "output_type": "execute_result"
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   "source": [
    "# nbi:hide_in\n",
    "\n",
    "def scattering_on_hard_sphere(b, R):\n",
    "    \n",
    "    x_min = -5\n",
    "    x_max = 3\n",
    "    y_min = -3\n",
    "    y_max = 5\n",
    "    points = 2\n",
    "    \n",
    "    if b<R:\n",
    "        theta = np.pi-2.*np.arcsin(b/R)\n",
    "        r_tp = np.sqrt(R**2 - b**2)\n",
    "        #line1:\n",
    "        x = np.linspace(x_min,-r_tp)\n",
    "        \n",
    "        #line2:\n",
    "        r = np.linspace(0.1,10.,points)\n",
    "        k = np.tan(theta)\n",
    "        if k<0:\n",
    "            x2 = np.linspace(-r_tp,x_min,points)\n",
    "        else:\n",
    "            x2 = np.linspace(-r_tp,x_max,points)\n",
    "        y2 = [k*i+b for i in  x2+r_tp]\n",
    "    else:\n",
    "        theta = np.pi\n",
    "        x = np.linspace(x_min,x_max, points)\n",
    "        x2 = y2 = np.zeros_like(x)\n",
    "    \n",
    "\n",
    "    y = np.zeros_like(x)\n",
    "    y[:] = b    \n",
    "        \n",
    "    #####PLOT\n",
    "    fig = plt.figure(figsize=(8, 8), dpi= 80, facecolor='w', edgecolor='k')\n",
    "    ax1 = fig.add_subplot()\n",
    "    ax1.axvline(0, color='gray')\n",
    "    ax1.axhline(0, color='gray')\n",
    "    \n",
    "    circle1 = plt.Circle((0, 0), R, color='#aec7df')\n",
    "    ax1.add_artist(circle1)\n",
    "    circle2 = plt.Circle((x_min+0.1, b), 0.1, color='#c83438')\n",
    "    ax1.add_artist(circle2)\n",
    "\n",
    "    ax1.plot(x,y,color='#c83438')\n",
    "    if b<=R:\n",
    "        ax1.plot(x2,y2,color='#c83438')\n",
    "    ax1.set_xlim(x_min,x_max)\n",
    "    ax1.set_ylim(y_min,y_max)\n",
    " \n",
    "    ax1.set_xlabel('y',fontsize=14)\n",
    "    ax1.set_ylabel('z',fontsize=14)    \n",
    "    plt.plot()\n",
    "    \n",
    "    \n",
    "interact(scattering_on_hard_sphere, b=(0.,3.), R=(0.,3.5))\n",
    "\n",
    "\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Scattering angle vs. impact parameter"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 26,
   "metadata": {},
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       "interactive(children=(FloatSlider(value=2.0, description='R', max=4.0), Output()), _dom_classes=('widget-inter…"
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    {
     "data": {
      "text/plain": [
       "<function __main__.theta_b_rigid(R)>"
      ]
     },
     "execution_count": 26,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# nbi:hide_in\n",
    "def theta_b_rigid(R):\n",
    "    \n",
    "    b = np.linspace(0,5,50)\n",
    "    theta = np.zeros_like(b)\n",
    "    for i,bb in enumerate(b):\n",
    "        if bb<R:\n",
    "            theta[i] = 2.*np.arccos(bb/R)\n",
    "        else:\n",
    "            theta[i] = 0.0\n",
    "            \n",
    "    if R==0.0:\n",
    "        theta[0] = np.pi\n",
    "\n",
    "    #####PLOT\n",
    "    fig = plt.figure(figsize=(8, 8), dpi= 80, facecolor='w', edgecolor='k')\n",
    "    ax1 = fig.add_subplot()\n",
    "\n",
    "    ax1.plot(b,theta,color='#c83438')\n",
    "    circle1 = plt.Circle((0, 0), R, color='#aec7df')\n",
    "    ax1.add_artist(circle1)\n",
    "    \n",
    "    ax1.set_xlim(0, np.pi)\n",
    "    ax1.set_ylim(0,np.pi)\n",
    "    ax1.set_ylabel(r'$\\theta$ / $\\pi$',fontsize=14)\n",
    "    ax1.set_xlabel('b',fontsize=14)    \n",
    "    plt.plot()\n",
    "\n",
    "interact(theta_b_rigid, R=(0.,4))\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Scattering of particles with Coulomb interaction\n",
    "\n",
    "Scattering angle:\n",
    "$$ \\theta(b) = \\pm2\\arccos{\\frac{1}{\\sqrt{\\gamma^2 + 1}}}   $$\n",
    "with\n",
    "$$ \\gamma = \\frac{C}{2Eb}  \\quad \\mathrm{where} C>0 $$\n",
    "\n",
    "Differential scattering cross section\n",
    "$$ \\frac{d\\sigma}{d\\Omega} =  \\left( \\frac{C}{4E} \\right)^2 \\frac{1}{\\sin^4{(\\theta/2)}} $$ \n",
    "\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "# nbi:hide_in\n",
    "\n",
    "def scattering_coulomb(b, R):\n",
    "    \n",
    "    x_min = -5\n",
    "    x_max = 3\n",
    "    y_min = -3\n",
    "    y_max = 5\n",
    "    points = 2\n",
    "    \n",
    "    if b<R:\n",
    "        theta = np.pi-2.*np.arcsin(b/R)\n",
    "        r_tp = np.sqrt(R**2 - b**2)\n",
    "        #line1:\n",
    "        x = np.linspace(x_min,-r_tp)\n",
    "        \n",
    "        #line2:\n",
    "        r = np.linspace(0.1,10.,points)\n",
    "        k = np.tan(theta)\n",
    "        if k<0:\n",
    "            x2 = np.linspace(-r_tp,x_min,points)\n",
    "        else:\n",
    "            x2 = np.linspace(-r_tp,x_max,points)\n",
    "        y2 = [k*i+b for i in  x2+r_tp]\n",
    "    else:\n",
    "        theta = np.pi\n",
    "        x = np.linspace(x_min,x_max, points)\n",
    "        x2 = y2 = np.zeros_like(x)\n",
    "    \n",
    "\n",
    "    y = np.zeros_like(x)\n",
    "    y[:] = b    \n",
    "        \n",
    "    #####PLOT\n",
    "    fig = plt.figure(figsize=(8, 8), dpi= 80, facecolor='w', edgecolor='k')\n",
    "    ax1 = fig.add_subplot()\n",
    "    ax1.axvline(0, color='gray')\n",
    "    ax1.axhline(0, color='gray')\n",
    "    \n",
    "    circle1 = plt.Circle((0, 0), R, color='#aec7df')\n",
    "    ax1.add_artist(circle1)\n",
    "    circle2 = plt.Circle((x_min+0.1, b), 0.1, color='#c83438')\n",
    "    ax1.add_artist(circle2)\n",
    "\n",
    "    ax1.plot(x,y,color='#c83438')\n",
    "    if b<=R:\n",
    "        ax1.plot(x2,y2,color='#c83438')\n",
    "    ax1.set_xlim(x_min,x_max)\n",
    "    ax1.set_ylim(y_min,y_max)\n",
    " \n",
    "    ax1.set_xlabel('y',fontsize=14)\n",
    "    ax1.set_ylabel('z',fontsize=14)    \n",
    "    plt.plot()\n",
    "    \n",
    "    \n",
    "interact(scattering_coulomb, b=(0.,3.), R=(0.,3.5))\n",
    "\n",
    "\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  }
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