Manually labeled equations in lattice vibration theory notebooks.

Author: Taylor-96

notebook/lattice-vibration/theory/theory_molecular_vibration.ipynb

@@ -34,10 +34,10 @@
     }
    ],
    "source": [
-    "%%javascript\n",
-    "MathJax.Hub.Config({\n",
-    "    TeX: { equationNumbers: { autoNumber: \"AMS\" } }\n",
-    "});"
+    "# %%javascript\n",
+    "# MathJax.Hub.Config({\n",
+    "#     TeX: { equationNumbers: { autoNumber: \"AMS\" } }\n",
+    "# });"
    ]
   },
   {
@@ -48,8 +48,7 @@
     "\n",
     "The Newton's equation of motion may be written in the Lagrangian formalism as :\n",
     "\\begin{equation}\n",
-    "    \\frac{d}{dt}\\frac{\\partial\\mathcal{L}}{\\partial \\dot{q_j}}-\\frac{\\partial \\mathcal{L}}{\\partial q_k}=0\n",
-    "    \\label{eq:Lagrangian}\n",
+    "    \\frac{d}{dt}\\frac{\\partial\\mathcal{L}}{\\partial \\dot{q_j}}-\\frac{\\partial \\mathcal{L}}{\\partial q_k}=0 \\qquad (1)\n",
     "\\end{equation}\n",
     "where\n",
     "\\begin{equation}\n",
@@ -64,17 +63,15 @@
     "<br>\n",
     "In mass-weighted coordinates, such as $q_{1}=\\sqrt{M_{1}} \\Delta x_{1}$, $q_{2}=\\sqrt{M_{1}} \\Delta y_{1}, q_{3}=\\sqrt{M_{1}} \\Delta z_{1}, q_{4}=\\sqrt{M_{2}} \\Delta x_{2}$, etc., the kinetic energy operator becomes simpler since the mass factors are now absorbed :\n",
     "\\begin{equation}\n",
-    "    K=\\frac{1}{2} \\sum_{i=1}^{3 N} \\dot{q}_{i}^{2}\n",
-    "    \\label{eq:kinetic}\n",
+    "    K=\\frac{1}{2} \\sum_{i=1}^{3 N} \\dot{q}_{i}^{2} \\qquad (2)\n",
     "\\end{equation}\n",
     "The potential energy ca be expended as:\n",
     "\\begin{equation}\n",
-    "    V=V_{0}+\\sum_{i=1}^{3 N}\\left(\\frac{\\partial V}{\\partial q_{i}}\\right)_{0} q_{i}+\\frac{1}{2} \\sum_{i=1}^{3 N}\\left(\\frac{\\partial^{2} V}{\\partial q_{i} \\partial q_{j}}\\right)_{0} q_{i} q_{j}+\\cdots\n",
-    "    \\label{eq:potential}\n",
+    "    V=V_{0}+\\sum_{i=1}^{3 N}\\left(\\frac{\\partial V}{\\partial q_{i}}\\right)_{0} q_{i}+\\frac{1}{2} \\sum_{i=1}^{3 N}\\left(\\frac{\\partial^{2} V}{\\partial q_{i} \\partial q_{j}}\\right)_{0} q_{i} q_{j}+\\cdots \\qquad (3)\n",
     "\\end{equation}\n",
     "At equilibrium $\\left.\\frac{\\partial V}{\\partial q_i}\\right|_{0}=0$. Since $V_0$ is arbitrary, it can be set to 0. For further readability, $\\left(\\partial^{2} V / \\partial q_{i} \\partial q_{j}\\right)_{0}$ will be abbreviated as $f_{i j}$. $f_{i j}$ can be interpreted as the force acting on atom $j$ from a small displacement of atom $i$.\n",
     "<br>\n",
-    "Eq. \\eqref{eq:kinetic} and Eq. \\eqref{eq:potential} are then inserted into Eq. \\eqref{eq:Lagrangian} :\n",
+    "Eq.2 and Eq.3 are then inserted into Eq.1 :\n",
     "\\begin{equation}\n",
     "    \\frac{d}{d t} \\frac{\\partial T}{\\partial \\dot{q}_{j}}+\\frac{\\partial V}{\\partial q_{j}}=0 \\quad j=1,2, \\cdots, 3 N\n",
     "\\end{equation}\n",

notebook/lattice-vibration/theory/theory_phonon_1d.ipynb

@@ -43,25 +43,22 @@
     "\n",
     "Let's now consider the case of a 1D monoatomic chain in which only the first neighbours interactions are considered. Let $R_n=na$ be the position of atom $n$, $R_{n+1}=(n+1)a$ the position of atom $n+1$, etc., as shown in Fig. 1. The classical equation of motion of the $n$-th atom of mass $M$ in position $R_n+u_n(t)$ under the force $F_n$ is:\n",
     "\\begin{equation}\n",
-    "    M\\ddot u_n = -C_1(u_{n-1}-u_{n})-C_1(u_{n+1}-u_n)=-C_1(2u_n-u_{n-1}-u_{n+1})\n",
-    "    \\label{eq:1d_eq}\n",
+    "    M\\ddot u_n = -C_1(u_{n-1}-u_{n})-C_1(u_{n+1}-u_n)=-C_1(2u_n-u_{n-1}-u_{n+1}) \\qquad (1)\n",
     "\\end{equation}\n",
     "where $n-1$ and $n+1$ are the two neighbouring atoms and $C_1$ the force constant between neighbours.<br>\n",
     "Solutions of the differential equation are in the form of traveling wave, periodic in space and time, of the type :\n",
     "\\begin{equation}\n",
-    "    u_n(t)=Ae^{i(k\\cdot R_n-\\omega t)}\n",
-    "    \\label{eq:u_1d}\n",
+    "    u_n(t)=Ae^{i(k\\cdot R_n-\\omega t)} \\qquad (2)\n",
     "\\end{equation}\n",
     "where $A$ is the amplitude of the displacement, $k$ is the phonon wave vector and $\\omega$ its frequency.<br>\n",
-    "Plugging Eq.\\ref{eq:u_1d} into Eq.\\eqref{eq:1d_eq} results in\n",
+    "Plugging Eq.2 into Eq.1results in\n",
     "$$\n",
     "\\begin{align}\n",
     "    -M \\omega^2 \\cancel{e^{ikn}}\\cancel{e^{-i\\omega t}} & =-C_1(2\\cancel{e^{ikn}}\\cancel{e^{-i\\omega t}}-\\cancel{e^{ikn}}\\cancel{e^{-i\\omega t}}e^{-ika}-\\cancel{e^{ikn}}\\cancel{e^{-i\\omega t}}e^{ika})\\nonumber \\\\\n",
-    "    M \\omega^2                                          & =-C_1(2-e^{-ika}-e^{ika}).\n",
-    "    \\label{eq:1d_final}\n",
+    "    M \\omega^2                                          & =-C_1(2-e^{-ika}-e^{ika}) \\qquad (3)\n",
     "\\end{align}\n",
     "$$\n",
-    "Eq. \\eqref{eq:1d_final} gives us a direct relation between $\\omega$ and $k$, which is called the dispersion relation."
+    "Eq.3 gives us a direct relation between $\\omega$ and $k$, which is called the dispersion relation."
    ]
   },
   {
@@ -99,16 +96,16 @@
    "source": [
     "## **1D diatomic chain**\n",
     "Let us now consider the case in which two atoms site in the unit cell, with mass $M_1$ and $M_2$. The positions are given by $R_n^{(1)}=na$ and $R_n^{(2)}=na+\\frac{1}{2}a$. $u$ is the displacement of atom 1 and $v$ is the displacement of atom 2. The system of equations is then\n",
+    "\n",
     "\\begin{align}\n",
-    "    M_1\\ddot u_n & =-C_1(2u_n-v_{n-1}-v_{n+1})\\label{eq:1} \\\\\n",
-    "    M_2\\ddot v_n & =-C_1(2v_n-u_{n-1}-u_{n+1})\\label{eq:2}\n",
+    "    M_1\\ddot u_n & =-C_1(2u_n-v_{n-1}-v_{n+1}) \\qquad (4) \\\\\n",
+    "    M_2\\ddot v_n & =-C_1(2v_n-u_{n-1}-u_{n+1})\\qquad (5)\n",
     "\\end{align}\n",
-    "The solution of such system is given by\n",
+    "The solution of such a system is given by\n",
     "\\begin{equation}\n",
-    "    u_n(t)=A_1e^{i(k\\cdot R_n^{(1)}-\\omega t)}\\qquad \\text{and}\\qquad  v_n(t)=A_2e^{i(k\\cdot R_n^{(2)}-\\omega t)}\n",
-    "    \\label{eq:3}\n",
+    "    u_n(t)=A_1e^{i(k\\cdot R_n^{(1)}-\\omega t)}\\qquad \\text{and}\\qquad  v_n(t)=A_2e^{i(k\\cdot R_n^{(2)}-\\omega t)} \\qquad(6)\n",
     "\\end{equation}\n",
-    "Replacing Eq.\\eqref{eq:3} into Eq.\\eqref{eq:1} and Eq.\\eqref{eq:2} gives :\n",
+    "Inserting Eq.6 into Eq.4 and Eq.5 gives :\n",
     "\\begin{align}\n",
     "     & -M_{1} \\omega^{2} A_{1}=-C\\left(2 A_{1}-A_{2} e^{-i k a / 2}-A_{2} e^{i k a / 2}\\right) \\\\\n",
     "     & -M_{2} \\omega^{2} A_{2}=-C\\left(2 A_{2}-A_{1} e^{-i k a / 2}-A_{1} e^{i k a / 2}\\right)\n",