diff --git a/notebook/lattice-vibration/theory/theory_molecular_vibration.ipynb b/notebook/lattice-vibration/theory/theory_molecular_vibration.ipynb index 5852fa3..75c011c 100644 --- a/notebook/lattice-vibration/theory/theory_molecular_vibration.ipynb +++ b/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 @@ "
\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", "
\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", diff --git a/notebook/lattice-vibration/theory/theory_phonon_1d.ipynb b/notebook/lattice-vibration/theory/theory_phonon_1d.ipynb index 19b73e3..28a1818 100644 --- a/notebook/lattice-vibration/theory/theory_phonon_1d.ipynb +++ b/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.
\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.
\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",