update LJ potential notebook

notebook/Lennard_Jones.ipynb

@@ -4,14 +4,71 @@
    "cell_type": "markdown",
    "metadata": {},
    "source": [
-    "# **Lennard Jones Potential**"
+    "# **Lennard-Jones Potential**\n",
+    "\n",
+    "<i class=\"fa fa-home fa-2x\"></i><a href=\"./index.ipynb\" style=\"font-size: 20px\"> Go back to index</a>\n",
+    "\n",
+    "**Source code:** https://github.com/osscar-org/materials-science/blob/master/notebook/Lennard_Jones.ipynb\n",
+    "\n",
+    "<hr style=\"height:1px;border:none;color:#cccccc;background-color:#cccccc;\" />"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "## **Goals**\n",
+    "\n",
+    "* We demonstrate the Lennard-Jones potential in this notebook.\n",
+    "    * Understand the interactions between particles and concept of the potentials.\n",
+    "    * Know the formula of the Lennard-Jones potential.\n",
+    "    * Understand the application of the Lennard-Jones potential in molecular modelling.\n",
+    "    * Know the insuffisance of the Lennard-Jones potential."
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "## **Background theory**\n",
+    "\n",
+    "In the molecular modelling, it is crucial to construct mathematical models for\n",
+    "the interactions of the simulated particles. Primarly, the simplest model should\n",
+    "be developed for two-body systems. The Lennard-Jones potential was proposed by\n",
+    "Sr Lennard Jones in 1931 \n",
+    "[[J E Lennard-Jones 1931 Proc. Phys. Soc. 43 461]](https://iopscience.iop.org/article/10.1088/0959-5309/43/5/301). The Lennard-Jones\n",
+    "potential is a simplifed and realistic model, which has proved successfully for \n",
+    "a varties of systems. Therefore, it is always used to be the first potential \n",
+    "model to teach students."
    ]
   },
   {
    "cell_type": "markdown",
    "metadata": {},
    "source": [
-    "[Go back to index](./index.ipynb)"
+    "<details close>\n",
+    "<summary style=\"font-size: 20px\">Lennard-Jones potential</summary>\n",
+    "The mathematical formula of the Lennard-Jones potential is:\n",
+    "    \n",
+    "$$\\large E(r)= \\epsilon \\left[\\frac{a}{r^{12}}-\\frac{b}{r^6}\\right]$$\n",
+    "    \n",
+    "where, $r$ is the distance between two atoms. $a$ and $b$ are two parameters.\n",
+    "$\\epsilon$ is the dispersion energy, which controls the depth of the\n",
+    "well potential. For a two-body system like two nuclei, the particles repulse \n",
+    "each other at short distance. While particles shows attraction at long distance.\n",
+    "The interaction should trend to vanlish at infinite distance.\n",
+    "The Lennard-Jones potential fulfills all these three conditions.\n",
+    "</details>"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "<details close>\n",
+    "<summary style=\"font-size: 20px\">Molecular simulations with LJ potential</summary>\n",
+    "\n",
+    "</details>"
    ]
   },
   {
@@ -20,30 +77,8 @@
    "source": [
     "<hr style=\"height:1px;border:none;color:#cccccc;background-color:#cccccc;\" />\n",
     "\n",
-    "<p style=\"text-align: justify;font-size:15px; width: 90%\"> \n",
-    "In this widget, it shows the Lennard-Jones potential. \n",
-    "The Lennard-Jones potential describes the interaction between atom, i.e. the energy depending on the distance. It is written as:\n",
-    "</p>\n",
-    "\n",
-    "<p style=\"text-align: justify;font-size:20px; width: 90%\"> \n",
-    "$E(r)= \\epsilon (\\frac{a}{r^12}-\\frac{b}{r^6})$\n",
-    "</p>\n",
-    "\n",
-    "<p style=\"text-align: justify;font-size:15px; width: 90%\">\n",
-    "where, $r$ is the distance between two atoms. $a$ and $b$ are two parameters. \n",
-    "It can be normalized by using $\\rho = \\frac{r_{min}}{r}$\n",
-    "The force can be written as:\n",
-    "</p>\n",
-    " \n",
-    "<p style=\"text-align: justify;font-size:20px; width: 90%\"> \n",
-    "$F(\\rho) = \\frac{d E}{d \\rho} = -12 \\rho(\\frac{1-\\rho^9}{\\rho^{14}})$\n",
-    "</p>\n",
-    "   \n",
-    "<p style=\"text-align: justify;font-size:15px; width: 90%\"> \n",
-    "The Wikipedia link of the Lennard-Jones potential:\n",
-    "</p>\n",
-    "\n",
-    "[https://en.wikipedia.org/wiki/Lennard-Jones_potential](https://en.wikipedia.org/wiki/Lennard-Jones_potential)"
+    "## Interactive visualization\n",
+    "(be patient, it might take a few seconds to load)"
    ]
   },
   {
@@ -189,7 +224,11 @@
     "p_dist2 = plt.scatter(x = [px2], y = [py2], colors =['orange'], stroke = 'black', default_size = 100)\n",
     "#p2 = plt.scatter(x = [1.0,], y = [0.0], colors =['orange'], stroke = 'black', default_size = 100)\n",
     "\n",
-    "display(VBox([HBox([fig1, fig2]), HBox([VBox([cb1, cb2]),VBox([cb3, cb4])]), a_slider, b_slider, HBox([dis, energy_button])]))"
+    "fig1.layout.min_width = \"400px\"\n",
+    "fig2.layout.min_width = \"400px\"\n",
+    "\n",
+    "display(HBox([VBox([fig1, cb1, cb2]), VBox([fig2, cb3, cb4])]))\n",
+    "display(a_slider, b_slider, HBox([dis, energy_button]))"
    ]
   },
   {
@@ -202,7 +241,7 @@
  ],
  "metadata": {
   "kernelspec": {
-   "display_name": "Python 3",
+   "display_name": "Python 3 (ipykernel)",
    "language": "python",
    "name": "python3"
   },
@@ -216,7 +255,7 @@
    "name": "python",
    "nbconvert_exporter": "python",
    "pygments_lexer": "ipython3",
-   "version": "3.7.8"
+   "version": "3.8.8"
   }
  },
  "nbformat": 4,

notebook/index.ipynb

@@ -25,7 +25,7 @@
  ],
  "metadata": {
   "kernelspec": {
-   "display_name": "Python 3",
+   "display_name": "Python 3 (ipykernel)",
    "language": "python",
    "name": "python3"
   },
@@ -39,7 +39,7 @@
    "name": "python",
    "nbconvert_exporter": "python",
    "pygments_lexer": "ipython3",
-   "version": "3.7.8"
+   "version": "3.8.8"
   }
  },
  "nbformat": 4,