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  {
   "cells": [
    {
     "cell_type": "heading",
     "level": 2,
     "metadata": {},
     "source": [
      "8. Consider the point group 4mm. It can be generated from a 4-fold rotation axis about the c-axis and a coinciding mirror plane about the a-c plane."
     ]
    },
    {
     "cell_type": "heading",
     "level": 2,
     "metadata": {},
     "source": [
      "a. Write down the 3D matrix representations for the two generators in the tetragonal basis."
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Solution:\n",
      "    \n",
      "The 4-fold rotation axis maps the $\\mathbf{a}$ and $\\mathbf{b}$ lattice vectors to $\\mathbf{a}'$ and $\\mathbf{b}'$, where $\\mathbf{a}'$ = $\\mathbf{b}$ and $\\mathbf{b}'$ = -$\\mathbf{a}$. Hence, the matrix is"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "D_4 = [[0, -1, 0],\n",
      "       [1, 0, 0],\n",
      "       [0, 0, 1]]"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 1
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "The mirror plane maps $\\mathbf{b}$ lattice vector to $\\mathbf{b}'$, where $\\mathbf{b}'$ = -$\\mathbf{b}$. Hence, the matrix is"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "D_m = [[1, 0, 0],\n",
      "       [0, -1, 0],\n",
      "       [0, 0, 1]]"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 2
    },
    {
     "cell_type": "heading",
     "level": 2,
     "metadata": {},
     "source": [
      "b. Determine the matrix representations for all symmetry elements in the group, and construct a multiplication table for the 4mm point group. "
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Solution: All symmetry elements can be obtained by the multiplication of the generators."
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "import numpy as np\n",
      "\n",
      "D = {}\n",
      "D[\"4\"] = np.array(D_4)\n",
      "D[\"m\"] = np.array(D_m)\n",
      "D[\"4_2\"] = np.dot(D[\"4\"], D[\"4\"])\n",
      "D[\"4_3\"] = np.dot(D[\"4\"], D[\"4_2\"])\n",
      "D[\"E\"] = np.dot(D[\"4\"], D[\"4_3\"])\n",
      "D[\"m_4\"] = np.dot(D[\"m\"], D[\"4\"])\n",
      "D[\"m_4_2\"] = np.dot(D[\"m\"], D[\"4_2\"])\n",
      "D[\"m_4_3\"] = np.dot(D[\"m\"], D[\"4_3\"])\n",
      "\n",
      "print \"The complete set of symmetry elements are given by the following matrices:\"\n",
      "for k, v in D.items():\n",
      "    print \"%s = %s\" % (k, str(v))"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "The complete set of symmetry elements are given by the following matrices:\n",
        "E = [[1 0 0]\n",
        " [0 1 0]\n",
        " [0 0 1]]\n",
        "m = [[ 1  0  0]\n",
        " [ 0 -1  0]\n",
        " [ 0  0  1]]\n",
        "4 = [[ 0 -1  0]\n",
        " [ 1  0  0]\n",
        " [ 0  0  1]]\n",
        "m_4_2 = [[-1  0  0]\n",
        " [ 0  1  0]\n",
        " [ 0  0  1]]\n",
        "m_4_3 = [[0 1 0]\n",
        " [1 0 0]\n",
        " [0 0 1]]\n",
        "4_2 = [[-1  0  0]\n",
        " [ 0 -1  0]\n",
        " [ 0  0  1]]\n",
        "4_3 = [[ 0  1  0]\n",
        " [-1  0  0]\n",
        " [ 0  0  1]]\n",
        "m_4 = [[ 0 -1  0]\n",
        " [-1  0  0]\n",
        " [ 0  0  1]]\n"
       ]
      }
     ],
     "prompt_number": 3
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "The multiplication table is given below:"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "import itertools\n",
      "keys = [\"E\", \"4\", \"4_2\", \"4_3\", \"m\", \"m_4\", \"m_4_2\", \"m_4_3\"]\n",
      "def find_key(m):\n",
      "    for k, v in D.items():\n",
      "        if np.all(v == m):\n",
      "            return k\n",
      "from prettytable import PrettyTable\n",
      "\n",
      "t = PrettyTable([\"\"] + keys)\n",
      "headers = list(keys)\n",
      "for k1 in keys:\n",
      "    row = [k1]\n",
      "    for k2 in keys:\n",
      "        prod = np.dot(D[k1], D[k2])\n",
      "        row.append(find_key(prod))\n",
      "    t.add_row(row)\n",
      "print t"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "+-------+-------+-------+-------+-------+-------+-------+-------+-------+\n",
        "|       |   E   |   4   |  4_2  |  4_3  |   m   |  m_4  | m_4_2 | m_4_3 |\n",
        "+-------+-------+-------+-------+-------+-------+-------+-------+-------+\n",
        "|   E   |   E   |   4   |  4_2  |  4_3  |   m   |  m_4  | m_4_2 | m_4_3 |\n",
        "|   4   |   4   |  4_2  |  4_3  |   E   | m_4_3 |   m   |  m_4  | m_4_2 |\n",
        "|  4_2  |  4_2  |  4_3  |   E   |   4   | m_4_2 | m_4_3 |   m   |  m_4  |\n",
        "|  4_3  |  4_3  |   E   |   4   |  4_2  |  m_4  | m_4_2 | m_4_3 |   m   |\n",
        "|   m   |   m   |  m_4  | m_4_2 | m_4_3 |   E   |   4   |  4_2  |  4_3  |\n",
        "|  m_4  |  m_4  | m_4_2 | m_4_3 |   m   |  4_3  |   E   |   4   |  4_2  |\n",
        "| m_4_2 | m_4_2 | m_4_3 |   m   |  m_4  |  4_2  |  4_3  |   E   |   4   |\n",
        "| m_4_3 | m_4_3 |   m   |  m_4  | m_4_2 |   4   |  4_2  |  4_3  |   E   |\n",
        "+-------+-------+-------+-------+-------+-------+-------+-------+-------+\n"
       ]
      }
     ],
     "prompt_number": 4
    },
    {
     "cell_type": "heading",
     "level": 2,
     "metadata": {},
     "source": [
      "c. What are the subgroups of the 4mm point group?"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Solution:\n",
      "\n",
      "4, m, m_4, m_4_2 and m_4_3 are subgroups of 4mm. "
     ]
    },
    {
     "cell_type": "heading",
     "level": 2,
     "metadata": {},
     "source": [
      "d. Draw a stereographic projection for the point group."
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Solutinon: \n",
      "    \n",
      "Refer to Structure of Materials Chapter 9, Fig 9.7."
     ]
    },
    {
     "cell_type": "heading",
     "level": 2,
     "metadata": {},
     "source": [
      "e. Determine the orbit for the general position (x, y, z)."
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Solution: \n",
      "    \n",
      "The orbit of the generalposition is given by:"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "from sympy import symbols\n",
      "x, y, z = symbols(\"x y z\")\n",
      "p = [x, y, z]\n",
      "for k, v in D.items():\n",
      "    print np.dot(v, p)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "[x y z]\n",
        "[x -y z]\n",
        "[-y x z]\n",
        "[-x y z]\n",
        "[y x z]\n",
        "[-x -y z]\n",
        "[y -x z]\n",
        "[-y -x z]\n"
       ]
      }
     ],
     "prompt_number": 5
    },
    {
     "cell_type": "heading",
     "level": 2,
     "metadata": {},
     "source": [
      "f. Identify possible special positions and determine the orbits of points lying on these special positions."
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Solution:\n",
      "\n",
      "There are special positions on the 4-fold axis (0, 0, z), the mirror plane (0, y, 0) and the diagonal mirror plane (x, x, z)."
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "The orbit for the special position (0, 0, z) on the 4-fold axis is given by:"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "p = [0, 0, z]\n",
      "orbit = []\n",
      "for k, v in D.items():\n",
      "    orbit.append(str(np.dot(v, p)))\n",
      "for o in set(orbit):\n",
      "    print o"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "[0 0 z]\n"
       ]
      }
     ],
     "prompt_number": 6
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "The orbit for the special position (0, y, 0) on the mirror plane on the a-c plane is given by:"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "p = [0, y, 0]\n",
      "orbit = []\n",
      "for k, v in D.items():\n",
      "    orbit.append(str(np.dot(v, p)))\n",
      "for o in set(orbit):\n",
      "    print o"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "[0 y 0]\n",
        "[0 -y 0]\n",
        "[-y 0 0]\n",
        "[y 0 0]\n"
       ]
      }
     ],
     "prompt_number": 7
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "The orbit for the special position (x, x, z) on the diagonal mirror plane is given by:"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "p = [x, x, z]\n",
      "orbit = []\n",
      "for k, v in D.items():\n",
      "    orbit.append(str(np.dot(v, p)))\n",
      "for o in set(orbit):\n",
      "    print o"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "[-x x z]\n",
        "[x x z]\n",
        "[-x -x z]\n",
        "[x -x z]\n"
       ]
      }
     ],
     "prompt_number": 8
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "**Note that all special positions have orbits that have < 8 elements.**"
     ]
    }
   ],
   "metadata": {}
  }
 ]
}