Heat_Capacity.ipynb

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  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Heat Capacity\n",
    "## Authors: Adrien Claret-Tournier and Bart Hoogenboom, University College London\n",
    "\n",
    "Here we explore how matter reacts at an atomic level to changes in temperature, and considering the individual and collective vibrations and taking into account quantum-mechanical effects.\n",
    "\n",
    "[Section 1](#1) **Solid State matter as a set of quantum harmonic oscillators**: We consider atoms in a solid as independent quantum harmonic oscillators.\n",
    "\n",
    "[Section 2](#2) **Wave Propagation: phonons and the heat capacity of solids**: We consider atoms in a solid as coupled oscillators, leading to collective excitations that we will consider as bosonic particles.\n",
    "\n",
    "### Running instructions\n",
    "\n",
    "The notebook runs from top to bottom and you should ``Run`` each input cell one after the other."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "If \"save = True\", the notebook outputs are saved. Note: In this case, the running time increases. \n",
    "\n",
    "If \"gif = True\", the file are saved as gif. Otherwise as mp4"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {},
   "outputs": [],
   "source": [
    "save = False # if true, it saves.\n",
    "gif = False # if true, it saves as gif. Otherwise, it saves as mp4."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [],
   "source": [
    "%%capture\n",
    "#### Import all necessary libraries, objects etc. ####\n",
    "# plotting\n",
    "import matplotlib as mpl\n",
    "import matplotlib.pyplot as plt\n",
    "import matplotlib.gridspec as gridspec\n",
    "\n",
    "# animation\n",
    "from matplotlib import animation\n",
    "from IPython.display import HTML\n",
    "!conda install -y -q ffmpeg tqdm # installs video reader \n",
    "from tqdm import tqdm_notebook as tqdm\n",
    "\n",
    "# interaction\n",
    "from ipywidgets import interact, interactive, fixed, interact_manual\n",
    "import ipywidgets as widgets\n",
    "\n",
    "# maths\n",
    "import numpy as np\n",
    "import random\n",
    "from scipy import integrate\n",
    "\n",
    "# Suppress runtime warnings\n",
    "import warnings\n",
    "warnings.filterwarnings('ignore')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [],
   "source": [
    "# Set font parameters for plots\n",
    "plt.rcParams[\"font.family\"] = \"Serif\"\n",
    "params = {'mathtext.default': 'regular' }\n",
    "plt.rcParams.update({'font.size': 14})"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 1) <a id='1'> </a> Solid state matter as a set of quantum harmonic oscillators \n",
    "\n",
    "We will consider atoms in a solid as a system of independent quantum harmonic oscillators. Each such oscillator has an average energy $\\langle E \\rangle$, which can be rewritten in terms of the partition function $Z$:\n",
    "\n",
    "$$\n",
    "\\langle E \\rangle = \\frac{1}{Z} \\sum_{q=0}^{\\infty} E_q e^{-\\beta E_q} =-\\frac{1}{Z}\\frac{\\partial Z}{\\partial \\beta} = -\\frac{\\partial \\ln Z}{\\partial \\beta} \\; ,\n",
    "$$\n",
    "\n",
    "where $E_q = (\\frac{1}{2} + q) \\hbar \\omega$ is the energy of a quantum harmonic oscillator with quantum number $q$, $\\omega$ is the (angular) oscillation frequency, and $\\hbar$ is Planck's constant. The partition function is found by summing the Boltzmann factors for all possible energies, leading to\n",
    "\n",
    "$$\n",
    "Z = e^{-\\frac{1}{2}\\beta \\hbar \\omega} \\sum_{q=0}^{\\infty} e^{-\\beta q \\hbar \\omega} = \\frac{1}{2\\sinh(\\frac{1}{2} \\beta \\hbar \\omega)} \\; .\n",
    "$$\n",
    "\n",
    "With this partition function, the average energy is\n",
    "\n",
    "$$\n",
    "\\langle E \\rangle = \\frac{\\hbar \\omega}{2\\tanh(\\frac{1}{2} \\beta \\hbar \\omega)} \\; .\n",
    "$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [],
   "source": [
    "%%capture\n",
    "# Init data for animation\n",
    "Nf = 30 # no. frames\n",
    "dt = 50 # frame time (ms)\n",
    "\n",
    "T = np.linspace(0.01,2.5, 100)\n",
    "\n",
    "# init average energy QHO\n",
    "def E(x):\n",
    "    return 0.5 / np.tanh(0.5/x)\n",
    "\n",
    "# init polynomial\n",
    "x = np.linspace(-1, 1,Nf)\n",
    "def y(a):\n",
    "    return a**2\n",
    "\n",
    "# calculate average of polynomial\n",
    "def avg(sc):\n",
    "    avg = 0\n",
    "    for i in x:\n",
    "        avg += y(sc*x)\n",
    "    avg = avg/len(x)\n",
    "    return avg\n",
    "\n",
    "# init temperatures and scaling factors for example energies\n",
    "T1 = 0.000001                      # chosen arbitrarily\n",
    "T2 = 0.5                      # \"\"\n",
    "T3 = 1                        # \"\"\n",
    "T4 = 2                        # \"\"\n",
    "\n",
    "#sc4 = 0.9\n",
    "#sc1 = sc4 * np.sqrt(T1 / T4)\n",
    "#sc2 = sc4 * np.sqrt(T2 / T4)\n",
    "#sc3 = sc4 * np.sqrt(T3 / T4)\n",
    "\n",
    "sc1 = 0.4\n",
    "sc2 = sc1 * np.sqrt(E(T2) / E(T1))\n",
    "sc3 = sc1 * np.sqrt(E(T3) / E(T1))\n",
    "sc4 = sc1 * np.sqrt(E(T4) / E(T1))\n",
    "\n",
    "# calculate position of ball depending on index i of animation (from 0 to Nf-1)     \n",
    "def move(sc, i):\n",
    "    # sorting out quadrants to map i between 0 and 199 to -1 and 1 (and back)\n",
    "    if (i < (Nf/4-1)):\n",
    "        px = sc * np.sin(np.pi/2 * (i / (Nf/4 - 1) - 1))\n",
    "    if (Nf/4 < i < (Nf/2 - 1)):\n",
    "        px = sc * np.sin(np.pi/2 * (i / (Nf/4) - 1) + np.pi/2)\n",
    "    if (Nf/2 < i < (3*Nf/2 - 1)):\n",
    "        px = sc * np.sin(np.pi/2 * ((i - Nf/2) / (Nf/4) + 1) + np.pi)\n",
    "    else:\n",
    "        px = sc * np.sin(np.pi/2 * (-(i - Nf/2) / (Nf/4) + 1) + np.pi)\n",
    "\n",
    "    py = 1.1*y(px) + 0.01 # y-coordinates\n",
    "    \n",
    "    pos = [-px,py] # direction correction\n",
    "    Ux.clear()\n",
    "    return pos\n",
    "\n",
    "\n",
    "# Set up figure and plot\n",
    "QHO = plt.figure(figsize=(12,6), constrained_layout=True)\n",
    "grid = QHO.add_gridspec(1, 2) # create a grid to arrange subplots\n",
    "Ux = QHO.add_subplot(grid[0,0]) # harmonic oscillator\n",
    "avgE = QHO.add_subplot(grid[0,1]) # average energy plot\n",
    "\n",
    "redLabel = \"$T_1=$\" + str.format('{0:0.1f}', T1)\n",
    "greenLabel = \"$T_2=$\" + str.format('{0:0.1f}', T2)\n",
    "blueLabel = \"$T_3=$\" + str.format('{0:0.1f}', T3)\n",
    "yellowLabel = \"$T_4=$\" + str.format('{0:0.1f}', T4)\n",
    "\n",
    "# Initialization function: plot the background of each frame\n",
    "def init():\n",
    "    Ux.clear()\n",
    "    \n",
    "    # Static graph: doesn't need to be called in animate()\n",
    "    avgE.clear()\n",
    "    avgE.set_xlim(0, 2.5)\n",
    "    avgE.set_ylim(0, 2.6)\n",
    "    avgE.set_xlabel('$k_B T/\\hbar\\omega$')\n",
    "    avgE.set_ylabel('$E / \\hbar \\omega$')\n",
    "    avgE.set_title('Average energy of the oscillator')\n",
    "    avgE.plot(T,E(T), label = 'Quantum harmonic oscillator')\n",
    "    avgE.plot(T, T,'--', label = 'Classical harmonic oscillator')\n",
    "    \n",
    "    avgE.plot(T1, E(T1), marker='o',color= 'r')\n",
    "    avgE.plot(T2, E(T2), marker='o',color= 'g')\n",
    "    avgE.plot(T3, E(T3), marker='o',color= 'b')\n",
    "    avgE.plot(T4, E(T4), marker='o',color= 'y')\n",
    "    avgE.legend()\n",
    "\n",
    "\n",
    "# Animation function which updates figure data\n",
    "def animate(i):\n",
    "    # calculate ball positions\n",
    "    pos1 = move(sc1, i)\n",
    "    pos2 = move(sc2, i)\n",
    "    pos3 = move(sc3, i)\n",
    "    pos4 = move(sc4, i)\n",
    "    \n",
    "    Ux.set_ylim(0,1)\n",
    "    Ux.tick_params(axis ='x', which ='both', length = 1)\n",
    "    Ux.yaxis.set_major_formatter(plt.NullFormatter())\n",
    "    Ux.set_xticks([0])\n",
    "    \n",
    "    # plot results\n",
    "    Ux.set_xlabel(\"Position\")\n",
    "    Ux.set_ylabel(\"Potential energy\")\n",
    "    Ux.set_title('Schematic of a quantum Harmonic oscillator')\n",
    "    Ux.plot(x, y(x))\n",
    "\n",
    "    \n",
    "    Ux.plot(pos1[0],pos1[1],'r.', label = redLabel, markersize = 20)\n",
    "    Ux.plot(pos2[0], pos2[1],'g.', label = blueLabel, markersize = 20)\n",
    "    Ux.plot(pos3[0], pos3[1], 'b.', label = greenLabel, markersize = 20)\n",
    "    Ux.plot(pos4[0], pos4[1], 'y.', label = yellowLabel, markersize = 20)\n",
    "    Ux.legend()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The following plots illustrate the dependence of the quantum harmonic oscillator on temperature.\n",
    "Left: Harmonic potential as a function of deviation of the equilibrium position. The oscillator is schematically indicated as a sphere moving in this potential, with an amplitude (= maximum deviation from equilibrium position) that scales with $\\sqrt{\\langle E \\rangle}$, shown for four different temperatures.\n",
    "Right: The average energy $\\langle E \\rangle$ os the quantum harmonic oscillator, compared with the prediction for the classical harmonic oscillator."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {
    "scrolled": false
   },
   "outputs": [
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     },
     "execution_count": 5,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Call the animator\n",
    "anim1 = animation.FuncAnimation(QHO, animate, init_func=init,frames=tqdm(range(Nf)), interval=dt)\n",
    "\n",
    "# Call function to display the animation\n",
    "HTML(anim1.to_html5_video())"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "The animation was not saved\n"
     ]
    }
   ],
   "source": [
    "# save the file\n",
    "if save == True:\n",
    "    \n",
    "    # save as gif\n",
    "    if gif == True:\n",
    "        writergif = animation.PillowWriter(fps=10) \n",
    "        anim1.save('energy_quantum_harmonic_oscillator.gif', writer=writergif)\n",
    "        print('The animation was saved as energy_quantum_harmonic_oscillator.gif')\n",
    "    \n",
    "    # save as mp4\n",
    "    else:\n",
    "        FFwriter = animation.FFMpegWriter(fps = 10) \n",
    "        anim1.save('energy_quantum_harmonic_oscillator.mp4', writer = FFwriter)\n",
    "        print('The animation was saved as energy_quantum_harmonic_oscillator.mp4')\n",
    "else:\n",
    "    print('The animation was not saved')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Calculating the heat capacity (Einstein model)\n",
    "\n",
    "If we consider a solid as a collection of $N$ oscillators that are independent (i.e., uncoupled) and that can each oscillate in 3 dimensions, we can write the average energy of the solid as the average energy of $3N$ quantum harmonic oscillators. That is,\n",
    "\n",
    "$$\n",
    "\\langle E \\rangle = 3N \\frac{\\hbar \\omega}{2\\tanh(\\frac{1}{2} \\beta \\hbar \\omega)} \\; .\n",
    "$$\n",
    "\n",
    "Hence we find the (Einstein) heat capacity,\n",
    "$$\n",
    "C = \\frac{\\text{d}\\langle E \\rangle}{\\text{d}T} = \\frac{3N \\hbar^2 \\omega_E^2} {4k_B T^2 \\sinh^2 (\\frac{\\hbar \\omega_E}{2k_B T})} \\; ,\n",
    "$$\n",
    "\n",
    "where we have defined the Einstein frequency, $\\omega_E \\equiv \\omega$, as the natural oscillation frequency of the atoms.\n",
    "\n",
    "This can be rewritten in terms of the Einstein temperature $T_E = \\hbar \\omega_E / k_B$, leading to\n",
    "$$\n",
    "C = \\frac{3N k_B T_E^2} {4 T^2 \\sinh^2 (\\frac{T_E}{2T})} \\; .\n",
    "$$\n",
    "\n",
    "Hereafter, we show the vibrations of the atoms in a (2D) lattice. As the temperature increases, the atoms store increasing amounts of energy in their vibrations, with the amplitude of the vibrations being proportional to $\\sqrt{\\langle E \\rangle}$. The heat capacity is defined by the additional uptake of vibrational energy as the temperature increases."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [],
   "source": [
    "%%capture\n",
    "# Define constants, data and functions relevant for both interaction and animation\n",
    "\n",
    "# Init data for animation\n",
    "Nf = 130 # no. frames\n",
    "dt = 75 # frame time (ms)\n",
    "\n",
    "# init data for plots\n",
    "# We will set N, k_B and hbar to 1, as the axis are in terms of these\n",
    "T = np.linspace(0.001,2.5, Nf)\n",
    "\n",
    "# Heat capacity\n",
    "def C(T):\n",
    "    return 3 / (4 * T**2 * np.sinh(0.5/T)**2)\n",
    "\n",
    "# vibration amplitude\n",
    "def vibAmp(T):\n",
    "    return np.sqrt(E(T))*0.02\n",
    "\n",
    "\n",
    "# init plots\n",
    "# Set up figure and plot\n",
    "CvPlots = plt.figure(figsize=(12,6), constrained_layout=True)\n",
    "grid = CvPlots.add_gridspec(1, 2) # create a grid to arrange subplots\n",
    "Cv = CvPlots.add_subplot(grid[0,1]) # heat capacity\n",
    "CvVib = CvPlots.add_subplot(grid[0,0]) # vibration\n",
    "\n",
    "n = 5\n",
    "\n",
    "# create arrays for particle positions\n",
    "xs = np.linspace(0.2, 0.8,n)\n",
    "ys = np.linspace(0.2,0.8,n)\n",
    "sine = np.linspace(0,18*2*np.pi, Nf) # particle oscillation positions\n",
    "\n",
    "# generate n random phases between 0 and 2pi\n",
    "def beats(n):\n",
    "    out = np.zeros(n)\n",
    "    for i in range(n):\n",
    "        out[i] = random.randint(0,100)/100*2*np.pi\n",
    "    \n",
    "    return out\n",
    "\n",
    "phases = beats(n*n)\n",
    "\n",
    "\n",
    "def init():\n",
    "    Cv.clear()\n",
    "    CvVib.clear()\n",
    "\n",
    "    \n",
    "def animate(frame):\n",
    "    Cv.clear()\n",
    "    CvVib.clear()\n",
    "    CvVib.set_xlim(0,1)\n",
    "    CvVib.set_ylim(0,1)\n",
    "    \n",
    "    Cv.set_xlabel(\"$T/T_E$\")\n",
    "    Cv.set_ylabel(\"$C_v/Nk_B$\")\n",
    "    Cv.set_title(\"Heat capacity as a function of temperature\")\n",
    "    CvVib.set_title(\"Lattice oscillations\")\n",
    "    \n",
    "    # calculate amplitude of oscillations and creates array\n",
    "    sc = vibAmp(T[frame])\n",
    "\n",
    "    # plot lattice\n",
    "    i=0\n",
    "    j=0\n",
    "    phasesCount=0\n",
    "    while i < n:\n",
    "        j=0\n",
    "        while j < n:\n",
    "            \n",
    "            xmove = sc * np.sin(sine[frame] + phases[phasesCount + j])\n",
    "            ymove = sc * np.sin(sine[frame] + phases[n*n - (phasesCount + j + 1)])\n",
    "            \n",
    "            CvVib.plot(xs[i] + xmove, ys[j] + ymove, 'b.') # build lattice\n",
    "            j+=1\n",
    "        i+=1\n",
    "        phasesCount += n\n",
    "\n",
    "    \n",
    "    \n",
    "    # Cv plot with animated movement\n",
    "    Cv.plot(T, C(T))\n",
    "    Cv.plot(T[frame], C(T[frame]), marker='o', color='r')\n",
    "    Cv.set_ylim(0,3)\n",
    "    Cv.set_xlim(0,2.5)\n",
    "    \n",
    "    # CvVib plot for independent beats\n",
    "    CvVib.tick_params(axis ='both', which ='both', length = 0)\n",
    "    CvVib.yaxis.set_major_formatter(plt.NullFormatter())\n",
    "    CvVib.xaxis.set_major_formatter(plt.NullFormatter())"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {
    "scrolled": false
   },
   "outputs": [
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       "</video>"
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      ]
     },
     "execution_count": 8,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Call the animator\n",
    "anim2 = animation.FuncAnimation(CvPlots, animate, init_func=init,frames=tqdm(range(Nf)), interval=dt)\n",
    "\n",
    "# Call function to display the animation\n",
    "HTML(anim2.to_html5_video())"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "As the red dot on the right climbs the heat capacity curve, the lattice vibrations become increasingly large, proportional to $\\sqrt{\\langle E\\rangle}$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "The animation was not saved\n"
     ]
    }
   ],
   "source": [
    "# save the file\n",
    "if save == True:\n",
    "    \n",
    "    # save as gif\n",
    "    if gif == True:\n",
    "        writergif = animation.PillowWriter(fps=10) \n",
    "        anim2.save('lattice_oscillation_heat_capacity.gif', writer=writergif)\n",
    "        print('The animation was saved as lattice_oscillation_heat_capacity.gif')\n",
    "    \n",
    "    # save as mp4\n",
    "    else:\n",
    "        FFwriter = animation.FFMpegWriter(fps = 10) \n",
    "        anim2.save('lattice_oscillation_heat_capacity.mp4', writer = FFwriter)\n",
    "        print('The animation was saved as lattice_oscillation_heat_capacity.mp4')\n",
    "else:\n",
    "    print('The animation was not saved')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## <a id='2'> </a>  2) Wave propagation: phonons and the heat capacity of solids\n",
    "\n",
    "In the Einstein model above, a solid is considered as $N$ independent 3D oscillators. However, quite generally, we may expect the position of an atom to be coupled to the atoms next to it. Due to such coupling, waves will form, which carry energy through the solid. Such waves can be longitudinal and transverse. The following code and animations show key concepts and steps in analysing the resulting collective vibrations, leading to the Debye model for the heat capacity of solids."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [],
   "source": [
    "%%capture\n",
    "#### Transverse and Longitudinal vibrations ####\n",
    "\n",
    "# init animation parameters\n",
    "Nf = 25 # no. frames\n",
    "dt = 40 # frame time (ms)\n",
    "\n",
    "# Set up figure and plot\n",
    "vibes = plt.figure(figsize=(13,7), constrained_layout=True)\n",
    "grid = vibes.add_gridspec(2, 1) # create a grid to arrange subplots\n",
    "long = vibes.add_subplot(grid[0,0]) # Longitudinal perturbation\n",
    "trans = vibes.add_subplot(grid[1,0]) # Transverse perturbation\n",
    "\n",
    "# Calculate displacement of each atom depending on the input mode to it\n",
    "def move(mode, frame, offI):\n",
    "    \n",
    "    offsets = np.sin(np.linspace(0,2*mode*np.pi, xparticles)) # creates starting pos for each particle depending on mode\n",
    "    \n",
    "    # creates all points that a particle will move to\n",
    "    if mode == 0:\n",
    "        oscillation = np.zeros(Nf)\n",
    "    else:\n",
    "        oscillation = np.sin(np.linspace(0 + offsets[offI],2*np.pi + offsets[offI], Nf)) \n",
    "    \n",
    "    displacement = 0.075 * oscillation[frame] # finds displacement for each particle\n",
    "    \n",
    "    return displacement\n",
    "\n",
    "\n",
    "# init values for 2D wave propagation\n",
    "xparticles = 20\n",
    "yparticles = 5\n",
    "\n",
    "\n",
    "# init point-building arrays\n",
    "x = np.linspace(0,1,xparticles)\n",
    "y = np.linspace(0,1,yparticles)\n",
    "zeros = np.zeros(yparticles)\n",
    "zeros2 = np.zeros(xparticles)\n",
    "\n",
    "\n",
    "# init animation frame etc\n",
    "def init():\n",
    "    long.clear()\n",
    "    trans.clear()\n",
    "    \n",
    "\n",
    "# Animation function which updates data\n",
    "def ani(frame):\n",
    "    long.clear()\n",
    "    trans.clear()\n",
    "       \n",
    "    long.set_xlim(-0.1,1.1)\n",
    "    long.set_ylim(-0.3, 1.3)\n",
    "    trans.set_xlim(-0.1,1.1)\n",
    "    trans.set_ylim(-0.3,1.3)\n",
    "    trans.set_title(\"Transverse wave\")\n",
    "    long.set_title(\"Longitudinal wave\")\n",
    "    \n",
    "    long.tick_params(axis ='both', which ='both', length = 0)\n",
    "    long.yaxis.set_major_formatter(plt.NullFormatter())\n",
    "    long.xaxis.set_major_formatter(plt.NullFormatter())\n",
    "    trans.tick_params(axis ='both', which ='both', length = 0)\n",
    "    trans.yaxis.set_major_formatter(plt.NullFormatter())\n",
    "    trans.xaxis.set_major_formatter(plt.NullFormatter())\n",
    "    \n",
    "    i=0\n",
    "    j=0\n",
    "    while j < xparticles:\n",
    "            \n",
    "        trans.plot(x[j] + zeros, y + move(mode, frame, j), 'b.')\n",
    "        long.plot(x[j] + zeros + 0.25 * move(mode, frame, j), y, 'b.')\n",
    "        \n",
    "        j += 1"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [
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     "output_type": "display_data"
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       "MA==\n",
       "\">\n",
       "  Your browser does not support the video tag.\n",
       "</video>"
      ],
      "text/plain": [
       "<IPython.core.display.HTML object>"
      ]
     },
     "execution_count": 11,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "choosemode = False # Set choosemode = True to allow user to input mode number\n",
    "if choosemode:\n",
    "    mode = int(input(\"Please enter mode (between 0-10) : \"))\n",
    "else:\n",
    "    mode = 2\n",
    "if mode < 0:\n",
    "    mode = 0\n",
    "    print(\"Out of bounds. Mode was set to 0.\")\n",
    "if mode > 10:\n",
    "    mode = 10\n",
    "    print(\"Out of bounds. Mode was set to 10.\")\n",
    "\n",
    "# Call the animator\n",
    "anim3 = animation.FuncAnimation(vibes, ani, init_func=init,frames=tqdm(range(Nf)), interval=dt)\n",
    "\n",
    "# Call function to display the animation\n",
    "HTML(anim3.to_html5_video())"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "The animation was not saved\n"
     ]
    }
   ],
   "source": [
    "# save the file\n",
    "if save == True:\n",
    "    \n",
    "    # save as gif\n",
    "    if gif == True:\n",
    "        writergif = animation.PillowWriter(fps=10) \n",
    "        anim3.save('transverse_longitudinal_waves.gif', writer=writergif)\n",
    "        print('The animation was saved as transverse_longitudinal_waves.gif')\n",
    "    \n",
    "    # save as mp4\n",
    "    else:\n",
    "        FFwriter = animation.FFMpegWriter(fps = 10) \n",
    "        anim3.save('transverse_longitudinal_waves.mp4', writer = FFwriter)\n",
    "        print('The animation was saved as transverse_longitudinal_waves.mp4')\n",
    "else:\n",
    "    print('The animation was not saved')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "To solve the equations of motion for coupled harmonic oscillators, we can make the Ansatz \n",
    "$$\n",
    "\\psi_j = \\sum_j e^{(kj - \\omega t)}\n",
    "$$\n",
    "for the displacement of atom $j$ away from its equilibrium position, where we consider only one dimension for simplicity. Here, we have defined the angular frequency of the wave $\\omega$, the wave number $k = n \\frac{2\\pi}{N} $, and $n = 1, 2,..., N$ is the mode of oscillation.\n",
    "\n",
    "Inserting this into the coupled equations of motion for the atoms, we find the dispersion relation,\n",
    "\n",
    "$$\n",
    "\\omega^2 = \\frac{4C_L}{m}\\sin^2\\left(\\frac{ka}{2}\\right) \\; ,\n",
    "$$\n",
    "\n",
    "where $C_L$ is the spring constant representing the elastic interaction between neighbouring atoms and $ka = \\frac{2\\pi}{\\lambda}$.\n",
    "\n",
    "Plotted below is the dispersion relation for arbitrary $\\lambda$ and $a$, with the mode of oscillation highlighted in red, and compared with the linear approximation that applies to a simple propagating wave, $\\omega^2 \\approx k^2 v_L^2$, with $v_L^2 = a^2 \\frac{C_L}{m}$. "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {
    "scrolled": false
   },
   "outputs": [
    {
     "data": {
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\n",
      "text/plain": [
       "<Figure size 1008x504 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "#### plotting dispersion relation ####\n",
    "\n",
    "# We will plot for C/m = 1 \n",
    "\n",
    "#lb = np.linspace(4,500,500)\n",
    "a = 1\n",
    "k = np.linspace(0, np.pi, 100)\n",
    "\n",
    "actualModes = np.linspace(0,np.pi, 11)\n",
    "\n",
    "w = np.sqrt(4 * np.sin(k/2)**2)\n",
    "wmode = np.sqrt(4 * np.sin(actualModes[mode]/2)**2)\n",
    "\n",
    "modeLabel = \"Oscillation mode $k = {}\\pi$\" .format(actualModes[mode]/np.pi)\n",
    "\n",
    "fig1 = plt.figure(figsize=(14,7))\n",
    "ax = fig1.add_subplot(111)\n",
    "\n",
    "ax.scatter(actualModes[mode], wmode, color = 'red', label = modeLabel, zorder = 2)\n",
    "ax.plot(k, w, label = \"Dispersion relation\", zorder = 1)\n",
    "ax.plot(k, k, '--', label = \"Linear approximation\", zorder = 0)\n",
    "\n",
    "ax.set_xticks(np.linspace(0, k[-1], 3))\n",
    "ax.set_xticklabels(['0', r'$\\frac{\\pi}{2 a}$', r'$\\frac{\\pi}{a}$'], fontsize ='large')\n",
    "ax.set_yticks(np.linspace(0, w[-1], 3))\n",
    "ax.set_yticklabels(['0', r'$\\frac{2 C_L}{m}$', r'$\\frac{4 C_L}{m}$'], fontsize ='large')\n",
    "\n",
    "ax.set_xlim(0,k[-1])\n",
    "ax.set_ylim(0,w[-1])\n",
    "ax.set_xlabel(\"$k$\")\n",
    "ax.set_ylabel(\"$\\omega$\")\n",
    "ax.set_title(\"Dispersion relation\")\n",
    "ax.legend();"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "The plot was not saved\n"
     ]
    }
   ],
   "source": [
    "if save == True:    \n",
    "    fig1.savefig('dispersion_relation.png')\n",
    "    print('The plot was saved as dispersion_relation.png')\n",
    "else:\n",
    "    print('The plot was not saved')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Each mode with angular frequency $\\omega$ may be associated with a bosonic particle with energy $\\hbar\\omega$. These bosonic particles are called phonons, and represent collective lattice vibrations. Each phonon has its own wave number $k$, and for small $k$, $\\omega \\approx kv$ ($v$: speed of sound in material). \n",
    "\n",
    "To determine the average energy contained by the lattice vibrations (and from there also the heat capacity), we first calculate the partition function $Z$:\n",
    "$$\n",
    "\\ln Z = \\int_0^{\\omega_D} g(\\omega) \\ln Z^{(\\omega)} \\text{d}\\omega \\;,\n",
    "$$\n",
    "where $Z^{(\\omega)}$ is the partition function of a single quantum harmonic oscillator with frequency $\\omega$. For small $k$ and $\\omega$, we use the (approximately) linear relation between $k$ and $\\omega$ to translate the density of states $\\rho (k) \\propto 4\\pi k^2$ into a density of states $g(\\omega) \\propto \\omega^2$ as a function of frequency. In addition, since there are $N$ atoms oscillating in 3D, there can be only $3N$ modes, so the integral over $g(\\omega) \\ln Z^{(\\omega)}$ must be bound, which is here done by capping it at $\\omega_D$, known as the Debye frequency. Specifically, we require that the integral over all modes $\\int_0^{\\omega_D} g(\\omega) \\text{d}\\omega = 3N$, the number of possible modes for $N$ particles vibrating in 3D.\n",
    "\n",
    "This leads to the Debye model for the heat capacity $C_v$,\n",
    "\n",
    "$$\n",
    "C_V = \\frac{\\text{d}\\langle E \\rangle}{\\text{d}T} = \\frac{3V \\hbar^2}{8 \\pi^2 v_s k_B T^2} \\int_0^{\\omega_D} \\frac{\\omega^4}{\\sinh^4(\\frac{1}{2}\\beta \\omega)} \\text{d}\\omega \\; .\n",
    "$$\n",
    "\n",
    "Rewriting this in terms of the Debye temperature $T_D = \\hbar \\omega_D / k_B$, we find\n",
    "\n",
    "$$\n",
    "C_V = 9V k_B \\left( \\frac{T}{T_D} \\right)^3 \\int_0^{T_D / T} \\frac{x^4 e^x}{(e^x - 1)^2} dx \\; ,\n",
    "$$\n",
    "\n",
    "with the low temperature limit\n",
    "\n",
    "$$\n",
    "C_V \\approx \\frac{12 \\pi^4}{5} Nk_B { \\left ( {\\frac{T}{T_D}} \\right ) }^3 \\; .\n",
    "$$\n",
    "\n",
    "\n",
    "These results may be compared with the Einstein model,\n",
    "$$ C_V = \\frac{3 N k_B T_E^2}{4 T^2 \\sinh^2 \\left ( \\frac{T_E}{2T} \\right )} \\; ,$$\n",
    "\n",
    "and with experimental data for the heat capacity of silver, as follows.\n",
    "\n",
    "Source of experimental data (here converted in appropriate units): J. Res. Natl. Inst. Stand. Technol. 100, 119 (1995), https://nvlpubs.nist.gov/nistpubs/jres/100/2/j12smi.pdf"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {},
   "outputs": [],
   "source": [
    "# Import data - will only work if file exists and path is correct; uncomment bottom 4 lines to load data from a file\n",
    "# Experimental data from J. Res. Natl. Inst. Stand. Technol. 100, 119 (1995), https://nvlpubs.nist.gov/nistpubs/jres/100/2/j12smi.pdf\n",
    "# data = np.genfromtxt(\"heat-capacity-of-silver-data.txt\",delimiter=\" \")\n",
    "# data = np.delete(data,range(2,7),1)\n",
    "# data[:,1]*=0.1078808\n",
    "# data[:,1]/=8.3145"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {},
   "outputs": [],
   "source": [
    "# Experimental data from J. Res. Natl. Inst. Stand. Technol. 100, 119 (1995), https://nvlpubs.nist.gov/nistpubs/jres/100/2/j12smi.pdf\n",
    "# C multiplied by 0.107880 and divided by 8.3145 to get in units of NkB\n",
    "exp_data = np.array([\n",
    "    [1.00000000e+00, 9.83506481e-05],[2.00000000e+00, 3.18796230e-04],[3.00000000e+00, 7.81874678e-04],\n",
    "    [4.00000000e+00, 1.61149743e-03],[5.00000000e+00, 2.92327190e-03],[6.00000000e+00, 4.84746730e-03],\n",
    "    [7.00000000e+00, 7.52940384e-03],[8.00000000e+00, 1.11507318e-02],[9.00000000e+00, 1.58814239e-02],\n",
    "    [1.00000000e+01, 2.20056331e-02],[1.10000000e+01, 2.97127948e-02],[1.20000000e+01, 3.90937339e-02],\n",
    "    [1.30000000e+01, 5.06415013e-02],[1.40000000e+01, 6.43560969e-02],[1.50000000e+01, 8.05878464e-02],\n",
    "    [1.60000000e+01, 9.93497246e-02],[1.70000000e+01, 1.20524957e-01],[1.80000000e+01, 1.44217342e-01],\n",
    "    [1.90000000e+01, 1.70076556e-01],[2.00000000e+01, 1.98102598e-01],[2.50000000e+01, 3.68788981e-01],\n",
    "    [3.00000000e+01, 5.74235440e-01],[3.50000000e+01, 7.95316798e-01],[4.00000000e+01, 1.01267433e+00],\n",
    "    [4.50000000e+01, 1.21607073e+00],[5.00000000e+01, 1.40246986e+00],[5.50000000e+01, 1.56855011e+00],\n",
    "    [6.00000000e+01, 1.71646534e+00],[6.50000000e+01, 1.84634528e+00],[7.00000000e+01, 1.96065520e+00],\n",
    "    [7.50000000e+01, 2.06160086e+00],[8.00000000e+01, 2.14944174e+00],[8.50000000e+01, 2.22884886e+00],\n",
    "    [9.00000000e+01, 2.29865446e+00],[9.50000000e+01, 2.36119406e+00],[1.00000000e+02, 2.41776514e+00],\n",
    "    [1.05000000e+02, 2.46823797e+00],[1.10000000e+02, 2.51274228e+00],[1.15000000e+02, 2.55361360e+00],\n",
    "    [1.20000000e+02, 2.59085190e+00],[1.25000000e+02, 2.62458695e+00],[1.30000000e+02, 2.65468900e+00],\n",
    "    [1.35000000e+02, 2.68349354e+00],[1.40000000e+02, 2.70879483e+00],[1.45000000e+02, 2.73279862e+00],\n",
    "    [1.50000000e+02, 2.75446690e+00],[1.55000000e+02, 2.77496743e+00],[1.60000000e+02, 2.79300271e+00],\n",
    "    [1.65000000e+02, 2.81103798e+00],[1.70000000e+02, 2.82660801e+00],[1.75000000e+02, 2.84230778e+00],\n",
    "    [1.80000000e+02, 2.85671005e+00],[1.85000000e+02, 2.86994457e+00],[1.90000000e+02, 2.88201134e+00],\n",
    "    [1.95000000e+02, 2.89407811e+00],[2.00000000e+02, 2.90601513e+00],[2.05000000e+02, 2.91561664e+00],\n",
    "    [2.10000000e+02, 2.92651566e+00],[2.15000000e+02, 2.93611717e+00],[2.20000000e+02, 2.94571869e+00],\n",
    "    [2.25000000e+02, 2.95415245e+00],[2.30000000e+02, 2.96258621e+00],[2.35000000e+02, 2.96985222e+00],\n",
    "    [2.40000000e+02, 2.97828599e+00],[2.45000000e+02, 2.98542225e+00],[2.50000000e+02, 2.99268826e+00],\n",
    "    [2.55000000e+02, 2.99865677e+00],[2.60000000e+02, 3.00592278e+00],[2.65000000e+02, 3.01189129e+00],\n",
    "    [2.70000000e+02, 3.01785980e+00],[2.73200000e+02, 3.02149280e+00],[2.75000000e+02, 3.02395805e+00],\n",
    "    [2.80000000e+02, 3.02992656e+00],[2.85000000e+02, 3.03602482e+00],[2.90000000e+02, 3.04082558e+00],\n",
    "    [2.95000000e+02, 3.04562634e+00],[2.98200000e+02, 3.04925934e+00],[3.00000000e+02, 3.05159485e+00]])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 864x864 with 3 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "Td = 215\n",
    "Te = 155\n",
    "# Te = Td * (np.pi/6)**(1/3) # ratio between Td and Te (https://en.wikipedia.org/wiki/Debye_model#Debye_versus_Einstein)\n",
    "T = np.arange(1, 300, 1) # temperatures from 0K to around room temperature (after which sinh approx is invalid)\n",
    "\n",
    "#exp_data = data #uncomment to retrieve data from file\n",
    "\n",
    "# Debye and Einstein models\n",
    "Ce = 3 / (4 * (T/Te)**2 * np.sinh(0.5*Te/T)**2) # Einstein model\n",
    "\n",
    "def Cdf(T, Td): # To calculate integral in Debye model\n",
    "    f = lambda x: x**4 * np.exp(x) / (np.exp(x) - 1)**2\n",
    "    \n",
    "    INT = integrate.quad(f, 0, Td/T)\n",
    "    value = 9*((T/Td)**3) * (INT[0] - INT[1])\n",
    "    \n",
    "    return value\n",
    "\n",
    "#heat capacity list for Debye model \n",
    "Cd = []\n",
    "i = 0\n",
    "l = len(T)\n",
    "while(i < l):\n",
    "    Cd.append(Cdf(T[i], Td))\n",
    "    i = i+ 1\n",
    "\n",
    "    \n",
    "# Set up figure and plot\n",
    "Plots = plt.figure(figsize=(12,12), constrained_layout=True)\n",
    "grid = Plots.add_gridspec(3, 2) # create a grid to arrange subplots\n",
    "DMod = Plots.add_subplot(grid[0,0]) # heat capacity\n",
    "EMod = Plots.add_subplot(grid[0,1]) # vibration\n",
    "comp = Plots.add_subplot(grid[1:,:]) # comparison\n",
    "\n",
    "comp.plot(T, Cd, color='r', label = \"Debye model\")\n",
    "comp.plot(T, Ce, color='b', label = \"Einstein model\")\n",
    "for i in range(len(exp_data)):\n",
    "    if i == 0:\n",
    "        comp.scatter(exp_data[i,0], exp_data[i,1], color = 'k', label = 'Experimental data', s = 10)\n",
    "    else:\n",
    "        comp.scatter(exp_data[i,0], exp_data[i,1], color = 'k',  s = 10)\n",
    "\n",
    "comp.set_xlim(0,300)\n",
    "comp.set_ylim(0,3* 1.1)\n",
    "comp.set_xlabel(\"$T/K$\")\n",
    "comp.set_ylabel(\"$C_V/Nk_B$\")\n",
    "comp.set_title(\"Einstein and Debye models compared\")\n",
    "comp.legend(loc=\"lower right\")\n",
    "\n",
    "DMod.plot(T/Td, Cd, label = \"$C_V$, Debye model\", color = 'r')\n",
    "DMod.set_title(\"Debye model\")\n",
    "DMod.set_xlabel(\"$T/T_D$\")\n",
    "DMod.set_ylabel(\"$C_V/Nk_B$\")\n",
    "DMod.set_xlim(0,300/230)\n",
    "DMod.set_ylim(comp.get_ylim())\n",
    "\n",
    "EMod.plot(T/Te, Ce, label = \"$C_V$, Einstein model\", color = 'b')\n",
    "EMod.set_title(\"Einstein model\")\n",
    "EMod.set_xlabel(\"$T/T_E$\")\n",
    "EMod.set_ylabel(\"$C_V/Nk_B$\")\n",
    "EMod.set_xlim(0,300/230)\n",
    "EMod.set_ylim(comp.get_ylim());"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "The plot was saved as einstein_debye_model.png\n"
     ]
    }
   ],
   "source": [
    "if save == True:    \n",
    "    Plots.savefig('einstein_debye_model.png')\n",
    "    print('The plot was saved as einstein_debye_model.png')\n",
    "else:\n",
    "    print('The plot was not saved')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "At low temperatures, the Debye model clearly matches the experimental data better than the Einstein model. In the high temperature limit, both models converge to 3 on the plot, consistent with the classical prediction based on the equipartition theorem, $C_V = 3Nk_B$."
   ]
  }
 ],
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   "display_name": "Python 3",
   "language": "python",
   "name": "python3"
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    "name": "ipython",
    "version": 3
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   "file_extension": ".py",
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   "nbconvert_exporter": "python",
   "pygments_lexer": "ipython3",
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