{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# NANO106 - Symmetry Computations on $m\\overline{3}m$ ($O_h$) Point Group\n",
    "*by Shyue Ping Ong*\n",
    "\n",
    "This notebook demonstrates the computation of orbits in the $m\\overline{3}m$ ($O_h$) point group (more complex than the simple mmm example). It is part of course material for UCSD's NANO106 - Crystallography of Materials."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Initializing the $m\\overline{3}m$ point group.\n",
    "\n",
    "Let's start by importing the numpy, sympy and other packages we need. Instead of going through all the steps one by one, we will use the symmetry.point_groups written by Prof Ong, which basically consolidates the information for all point groups in a single module."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "The generators for this point group are:\n",
      "[[0, 0, 1], [1, 0, 0], [0, 1, 0]]\n",
      "[[0, -1, 0], [1, 0, 0], [0, 0, 1]]\n",
      "[[-1, 0, 0], [0, -1, 0], [0, 0, -1]]\n",
      "[[0, 1, 0], [1, 0, 0], [0, 0, 1]]\n",
      "The order of the point group is 48.\n"
     ]
    }
   ],
   "source": [
    "import numpy as np\n",
    "from sympy import symbols, Mod\n",
    "from symmetry.groups import PointGroup\n",
    "\n",
    "#Create the point group.\n",
    "oh = PointGroup(\"m-3m\")\n",
    "\n",
    "print \"The generators for this point group are:\"\n",
    "for m in oh.generators:\n",
    "    print m\n",
    "print \"The order of the point group is %d.\" % len(oh.symmetry_ops)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Computing Orbits\n",
    "\n",
    "Using sympy, we specify the symbolic symbols x, y, z to represent position coordinates. We also define a function to generate the orbit given a set of symmetry operations and a point p."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "x, y, z = symbols(\"x y z\")\n",
    "\n",
    "def pt_2_str(pt):\n",
    "    return str([i.args[0] if isinstance(i, Mod) else i for i in pt])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Orbit for General Position"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "For the general position [x y z] on the two-fold axis, the orbit comprise 48 points:\n",
      "[z, x, y] [-y, x, z] [-x, -y, -z] [y, x, z] [y, z, x] [z, -y, x] [-z, -x, -y] [z, y, x] [-x, z, y] [-x, -y, z] [y, -x, -z] [-x, y, z] [x, y, z] [-y, -x, -z] [x, z, y] [x, -y, z] [x, z, -y] [-y, -z, -x] [z, y, -x] [-y, z, -x] [-x, -z, y] [y, z, -x] [-x, -z, -y] [-y, z, x] [y, -x, z] [z, -x, -y] [-z, y, -x] [z, -x, y] [x, -z, y] [-y, x, -z] [y, -z, -x] [-z, x, y] [z, -y, -x] [-z, x, -y] [y, -z, x] [-z, -y, -x] [-z, -x, y] [z, x, -y] [-z, y, x] [x, -z, -y] [-x, z, -y] [x, y, -z] [-y, -x, z] [-y, -z, x] [-z, -y, x] [y, x, -z] [x, -y, -z] [-x, y, -z]\n"
     ]
    }
   ],
   "source": [
    "p = np.array([x, y, z])\n",
    "orb = oh.get_orbit(p, tol=0)\n",
    "print \"For the general position %s on the two-fold axis, the orbit comprise %d points:\" % (str(p), len(orb))\n",
    "for o in orb:\n",
    "    print pt_2_str(o),"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Orbit for Special Position on four-fold rotation axis"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "For the special position [0 0 z] on the two-fold axis, the orbit comprise 6 points:\n",
      "[z, 0, 0] [0, 0, z] [0, 0, -z] [0, z, 0] [-z, 0, 0] [0, -z, 0]\n"
     ]
    }
   ],
   "source": [
    "p = np.array([0, 0, z])\n",
    "orb = oh.get_orbit(p, tol=0)\n",
    "\n",
    "print \"For the special position %s on the two-fold axis, the orbit comprise %d points:\" % (str(p), len(orb))\n",
    "for o in orb:\n",
    "    print pt_2_str(o),"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The orbit is similar for the other two-fold axes on the a and b axes are similar."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Orbit for Special Position on three-fold rotation axis\n",
    "\n",
    "The three-fold rotation axis are given by (x, x, x)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "For the special position [x x x] on the two-fold axis, the orbit comprise 8 points:\n",
      "[x, x, x] [-x, x, x] [-x, -x, -x] [x, -x, x] [-x, -x, x] [x, -x, -x] [x, x, -x] [-x, x, -x]\n"
     ]
    }
   ],
   "source": [
    "p = np.array([x, x, x])\n",
    "orb = oh.get_orbit(p, tol=0)\n",
    "print \"For the special position %s on the two-fold axis, the orbit comprise %d points:\" % (str(p), len(orb))\n",
    "for o in orb:\n",
    "    print pt_2_str(o),"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Orbit for Special Position on two-fold rotation axis\n",
    "\n",
    "The two-fold rotation axis are given by (x, x, 0)."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "For the special position [x x 0] on the two-fold axis, the orbit comprise 12 points:\n",
      "[0, x, x] [-x, x, 0] [-x, -x, 0] [x, x, 0] [x, 0, x] [0, -x, x] [0, -x, -x] [-x, 0, x] [x, -x, 0] [x, 0, -x] [-x, 0, -x] [0, x, -x]\n"
     ]
    }
   ],
   "source": [
    "p = np.array([x, x, 0])\n",
    "orb = oh.get_orbit(p, tol=0)\n",
    "print \"For the special position %s on the two-fold axis, the orbit comprise %d points:\" % (str(p), len(orb))\n",
    "for o in orb:\n",
    "    print pt_2_str(o),"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Orbit for Special Position on mirror planes\n",
    "\n",
    "Positions on the mirror on the a-b plane have coordinates (x, y, 0)."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "For the special position [x y 0] on the two-fold axis, the orbit comprise 24 points:\n",
      "[0, x, y] [-y, x, 0] [-x, -y, 0] [y, x, 0] [y, 0, x] [0, -y, x] [0, -x, -y] [0, y, x] [-x, 0, y] [y, -x, 0] [-x, y, 0] [x, y, 0] [-y, -x, 0] [x, 0, y] [x, -y, 0] [x, 0, -y] [-y, 0, -x] [0, y, -x] [y, 0, -x] [-x, 0, -y] [-y, 0, x] [0, -x, y] [0, -y, -x] [0, x, -y]\n"
     ]
    }
   ],
   "source": [
    "p = np.array([x, y, 0])\n",
    "orb = oh.get_orbit(p, tol=0)\n",
    "print \"For the special position %s on the two-fold axis, the orbit comprise %d points:\" % (str(p), len(orb))\n",
    "for o in orb:\n",
    "    print pt_2_str(o),"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The orbit is similar for the other mirror planes on the a-c and b-c planes are similar."
   ]
  }
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