CDML.ipynb

{
 "cells": [
  {
   "cell_type": "code",
   "execution_count": 1,
   "id": "5f9e7677",
   "metadata": {},
   "outputs": [],
   "source": [
    "import numpy as np\n",
    "from sympy import *\n",
    "init_printing()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "id": "68039e4f",
   "metadata": {},
   "outputs": [],
   "source": [
    "# Table 3.3 of CDML\n",
    "# For column-wise primitive basis vectors w.r.t. conventional lattice\n",
    "P_A = Matrix([\n",
    "    [1, 0, 0],\n",
    "    [0, 1/2, -1/2],\n",
    "    [0, 1/2, 1/2],\n",
    "])\n",
    "P_C = Matrix([\n",
    "    [1/2, 1/2, 0],\n",
    "    [-1/2, 1/2, 0],\n",
    "    [0, 0, 1],\n",
    "])\n",
    "P_F = Matrix([\n",
    "    [0, 1/2, 1/2],\n",
    "    [1/2, 0, 1/2],\n",
    "    [1/2, 1/2, 0],\n",
    "])\n",
    "P_I = Matrix([\n",
    "    [-1/2, 1/2, 1/2],\n",
    "    [1/2, -1/2, 1/2],\n",
    "    [1/2, 1/2, -1/2],\n",
    "])\n",
    "P_R_reverse = Matrix([\n",
    "    [1/3, 1/3, -2/3],\n",
    "    [-1/3,2/3, -1/3],\n",
    "    [1/3, 1/3, 1/3],\n",
    "])\n",
    "P_R_obverse = Matrix([\n",
    "    [2/3, -1/3, -1/3],\n",
    "    [1/3, 1/3, -2/3],\n",
    "    [1/3, 1/3, 1/3],\n",
    "])\n",
    "\n",
    "transformations = {\n",
    "    'mA_oA': P_A,\n",
    "    'mC_oC': P_C,\n",
    "    'oF_cF': P_F,\n",
    "    'oI_cI': P_I,\n",
    "    'tI': P_I,\n",
    "    'hP': Matrix([[1, 0, 0], [0, 1, 0], [0, 0, 1]]),\n",
    "    'hR_CDML': P_R_reverse,\n",
    "    'hR_spglib': P_R_obverse,\n",
    "}"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "id": "37b6a459",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "mA_oA 0.500000000000000\n",
      "mC_oC 0.500000000000000\n",
      "oF_cF 0.250000000000000\n",
      "oI_cI 0.500000000000000\n",
      "tI 0.500000000000000\n",
      "hP 1\n",
      "hR_CDML 0.333333333333333\n",
      "hR_spglib 0.333333333333333\n"
     ]
    }
   ],
   "source": [
    "for name, P in transformations.items():\n",
    "    print(name, P.det())"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "880106a8",
   "metadata": {},
   "source": [
    "When basis vectors are transformed from $A_{c}=(a_1, a_2, a_3)$ to $A_{p} := A_{c}P$, its dual basis vectors are transformed from $B_{c} := (b_1, b_2, b_3) = A^{-\\top}$ to $B_{p} := BP^{-\\top}$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "id": "8ad277e0",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "mA_oA\n"
     ]
    },
    {
     "data": {
      "image/png": "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",
      "text/latex": [
       "$\\displaystyle \\left[\\begin{matrix}k_{1} & k_{2} - k_{3} & k_{2} + k_{3}\\end{matrix}\\right]$"
      ],
      "text/plain": [
       "[k₁  k₂ - k₃  k₂ + k₃]"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "mC_oC\n"
     ]
    },
    {
     "data": {
      "image/png": "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",
      "text/latex": [
       "$\\displaystyle \\left[\\begin{matrix}k_{1} + k_{2} & - k_{1} + k_{2} & k_{3}\\end{matrix}\\right]$"
      ],
      "text/plain": [
       "[k₁ + k₂  -k₁ + k₂  k₃]"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "oF_cF\n"
     ]
    },
    {
     "data": {
      "image/png": "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",
      "text/latex": [
       "$\\displaystyle \\left[\\begin{matrix}- k_{1} + k_{2} + k_{3} & k_{1} - k_{2} + k_{3} & k_{1} + k_{2} - k_{3}\\end{matrix}\\right]$"
      ],
      "text/plain": [
       "[-k₁ + k₂ + k₃  k₁ - k₂ + k₃  k₁ + k₂ - k₃]"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "oI_cI\n"
     ]
    },
    {
     "data": {
      "image/png": "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",
      "text/latex": [
       "$\\displaystyle \\left[\\begin{matrix}k_{2} + k_{3} & k_{1} + k_{3} & k_{1} + k_{2}\\end{matrix}\\right]$"
      ],
      "text/plain": [
       "[k₂ + k₃  k₁ + k₃  k₁ + k₂]"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "tI\n"
     ]
    },
    {
     "data": {
      "image/png": "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",
      "text/latex": [
       "$\\displaystyle \\left[\\begin{matrix}k_{2} + k_{3} & k_{1} + k_{3} & k_{1} + k_{2}\\end{matrix}\\right]$"
      ],
      "text/plain": [
       "[k₂ + k₃  k₁ + k₃  k₁ + k₂]"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "hP\n"
     ]
    },
    {
     "data": {
      "image/png": "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",
      "text/latex": [
       "$\\displaystyle \\left[\\begin{matrix}k_{1} & k_{2} & k_{3}\\end{matrix}\\right]$"
      ],
      "text/plain": [
       "[k₁  k₂  k₃]"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "hR_CDML\n"
     ]
    },
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAATYAAAAZCAYAAAC7MQL+AAAACXBIWXMAAA7EAAAOxAGVKw4bAAAFTklEQVR4Ae1d2VEcMRBdKALAOAOcgcERYDIwdgSGDNjiC/4oyMDOwEcGxhkAGRhHALUh+L1B2tKKZUY9O9KocXeVrGN0vH6SenTs4LXT09Mfk8nkLZyX/bOzszsfMd8YMAaMgZoZgL06Br6jAON0DYbtBg92gkQLGgPGgDGgkgHYskMCX1eJ3kAbA8aAMdDCwEbLswms3yaec6u6C/eA+Ju2/DU804hZyptWHTXg1oAxHi+GOWakY8UGwmZw+yjGM7erp8XrS9GIWcqiVh014NaAMR4vhjlmpMOwBdl5ufAriGsIasQs5bWojphA7+GaMwwp0Ch/UdxR26nRYhi18joQ7iw8d56xEbwbCSpWbMSqEXPqbPP5RtKRRxN0vWUk3CK8I2DUyutKuHPy3GnYMCK4FeWWdCYaHeNm1ohZyphWHTXg1oAxHi+GOWCk9fLA5eOKbb5ag4H7gPg7OFrraaUGTyNm0CkSrTpqwK0BYzxYDHPASOuKDUaLxmu+B3ZGjcW/wH10jvFqRCNmKXladdSAWwPGeLwY5piR7ssDvgUoVyCvCcP/iTgNXq03pRoxk2OJaNVRA24NGOOxYpgjRjaieBxt9u1IJHHXMGrNlhT+LeILXysgjcaOq7gjhBeeIe2JuPy/8YDlUuXAtd2WPwkz6tlGJdTrAY5hbq/PE+pHtv4ykN5JOhKlay+5X1wZrsj9ZGGSly0GUGf4+Yp/dov0Ax95xk/C7TCfuDrYN5TPSJ89BrP+m4SRCBzOZG6RXyWvmXAn8Yy2e83TLsPGwc2VGQ3VDhWEo1FbEKRxu+onQpKhQhkO0k4DuNBQWiQJM6q6gLsDjq+sFv4xPBraV4znkoH0TtIRbYn7hXqj3DLDxXSer27Dv+zJTxJu1H0RYkCYBuEGrsQPxJMwApOY21CnkD+kV81rJtxJPHMswInn6XpIcBiGMjRQtJZcxXCg8wsEflfKDl0QpPFtzcE+6sfzwJCMGVjP4b4FirxGmKu3qkWiYy39QkIluJH9EPk58L1wcNOgPhl7PsMQvgQj8moc86PwGveNhGeU7TVPnzVsqJBLbIq/Eb1+jDbGbgJwXOHUJsmY3cAMV598a3IC1S7JOlamiAQ3X6R+vJVUQ4KxJK62tiSYx+I1xp+Mue883YhbDOLcA/OtxC1jKH5VxhVObSLG7Az0JyjCbXazLa1NqQiPWMeo/FjRZNxL+oETktuR8EWUQ49kjDka71lnMuYReY1VS8bsCwL7McLJ87RtxcZtHZeBjaBiGrgpHC8H2Mj8WZOhjn/EmKELt9B7cPw7dNSrdhHrWIlCvXCjT7j95Go6x3lsTE0vjHElheO9MBfmNaZEjFk6Twf9e2xovNnOwS9xyBuTtVLcdTQPqHlJkntlsBJWaeGh+sXVs8rlgQg62uMZLy8OeBs+ExUulHkIbrXyWho3uxRt8kX37DzF8+Zb5rataKGhUb4ZKM83xl+4PYS9EfMTZxfpPq08uLpbJEeep6xI0S80avyyhduWiYvT90chWdsvXLlWXrPiRl9voh96zdP/1bDx29cHkBZOEt7CsaO+w5ksYQCc+YukJU+HS0I7fqVGw+ZvQnnOxqOQFydaec2NG/X3nqeDGDYA4EDkwKNx4FaFt4v38Pv+3gnFswt/THoCjPeuJf5Al9tQGrcXIdBFY7+Qe241+LamPxfowzFWhSjltnpel3Rur3k66BnbElCWZAwYA8ZAMQbwwrH/86AY29aQMWAMFGVgvWhr1pgxYAwYAwUYaM7YsHz7E7Rl/69oQIYFjQFjoG4GYL/4+1N//rqF8PQfzYZ6zxVtvcgAAAAASUVORK5CYII=",
      "text/latex": [
       "$\\displaystyle \\left[\\begin{matrix}k_{1} - k_{3} & - k_{1} + k_{2} & k_{1} + k_{2} + k_{3}\\end{matrix}\\right]$"
      ],
      "text/plain": [
       "[k₁ - k₃  -k₁ + k₂  k₁ + k₂ + k₃]"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "hR_spglib\n"
     ]
    },
    {
     "data": {
      "image/png": "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",
      "text/latex": [
       "$\\displaystyle \\left[\\begin{matrix}k_{1} - k_{2} & k_{2} - k_{3} & k_{1} + k_{2} + k_{3}\\end{matrix}\\right]$"
      ],
      "text/plain": [
       "[k₁ - k₂  k₂ - k₃  k₁ + k₂ + k₃]"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "# Reproduce Table 2 of Aroyo et al., (2014)\n",
    "# hR(reverse) result is not consistent with the table!\n",
    "conventional = Matrix([symbols('k_1'), symbols('k_2'), symbols('k_3')])\n",
    "for name, P in transformations.items():\n",
    "    print(name)\n",
    "    PinvT = Matrix(np.around(np.linalg.inv(np.array(P, dtype=float)).T).astype(int))\n",
    "    display((PinvT * conventional).T)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "13270ff4",
   "metadata": {},
   "source": [
    "Real lattice oI (cI) corresponds to reciprocal lattice oF (cF) by taking dual of primitive basis vectors.\n",
    "However, dual of conventional basis of oI (cI) is not conventional reciprocal basis of oF (cF).\n",
    "The adjusted reciprocal basis vectors is obtained by\n",
    "$$\n",
    "    B_{a}(oI, cI) = B_{p}(oI, cI)\n",
    "                    \\begin{pmatrix}\n",
    "                        -1 & 1 & 1 \\\\\n",
    "                        1 & -1 & 1 \\\\\n",
    "                        1 & 1 & -1 \\\\\n",
    "                    \\end{pmatrix} \\\\\n",
    "    B_{a}(oF, cF) = B_{p}(oF, cF)\n",
    "                    \\begin{pmatrix}\n",
    "                        0 & 1 & 1 \\\\\n",
    "                        1 & 0 & 1 \\\\\n",
    "                        1 & 1 & 0 \\\\\n",
    "                    \\end{pmatrix}\n",
    "$$\n",
    "\n",
    "$$\n",
    "    \\mathbf{k}_{c} = P^{-\\top} \\mathbf{k}_{p}\n",
    "        \\quad \\mathrm{(Table~2)} \\\\\n",
    "    \\mathbf{k}_{p} := Q \\mathbf{k}_{a}\n",
    "        \\quad \\mathrm{(Table~1.5.4.2)} \\\\\n",
    "    \\mathbf{k}_{c} = P^{-\\top} Q \\mathbf{k}_{a}\n",
    "        \\quad \\mathrm{(Table~1.5.4.1)}\n",
    "$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "id": "267963e2",
   "metadata": {},
   "outputs": [],
   "source": [
    "adjusted = {\n",
    "    'mA_oA': Matrix([\n",
    "        [1, 0, 0],\n",
    "        [0, 1, -1],\n",
    "        [0, 1, 1],\n",
    "    ]),\n",
    "    'mC_oC': P_C.inv(),\n",
    "    'oF_cF': P_I.inv(),  # Use I-centered!\n",
    "    'oI_cI': P_F.inv(),  # Use F-centered!\n",
    "    'tI': Matrix([\n",
    "        [-1, 0, 1],\n",
    "        [1, 0, 1],\n",
    "        [0, 1, -1],\n",
    "    ]),\n",
    "    'hP': Matrix([\n",
    "        [1, -1, 0],\n",
    "        [0, 1, 0],\n",
    "        [0, 0, 1],\n",
    "    ]),\n",
    "    'hR_CDML': P_R_reverse.inv(),  # TODO\n",
    "    'hR_spglib': P_R_obverse.inv(),  # TODO\n",
    "}"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "id": "ce6895cc",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "mA_oA\n"
     ]
    },
    {
     "data": {
      "image/png": "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",
      "text/latex": [
       "$\\displaystyle \\left[\\begin{matrix}k_{a1} & k_{a2} - k_{a3} & k_{a2} + k_{a3}\\end{matrix}\\right]$"
      ],
      "text/plain": [
       "[k_{a1}  k_{a2} - k_{a3}  k_{a2} + k_{a3}]"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "mC_oC\n"
     ]
    },
    {
     "data": {
      "image/png": "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",
      "text/latex": [
       "$\\displaystyle \\left[\\begin{matrix}k_{a1} - k_{a2} & k_{a1} + k_{a2} & k_{a3}\\end{matrix}\\right]$"
      ],
      "text/plain": [
       "[k_{a1} - k_{a2}  k_{a1} + k_{a2}  k_{a3}]"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "oF_cF\n"
     ]
    },
    {
     "data": {
      "image/png": "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",
      "text/latex": [
       "$\\displaystyle \\left[\\begin{matrix}k_{a2} + k_{a3} & k_{a1} + k_{a3} & k_{a1} + k_{a2}\\end{matrix}\\right]$"
      ],
      "text/plain": [
       "[k_{a2} + k_{a3}  k_{a1} + k_{a3}  k_{a1} + k_{a2}]"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "oI_cI\n"
     ]
    },
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAeIAAAAZCAYAAADtyy3NAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAHCUlEQVR4Ae2d73HcNhDFJY0KUJwOnA5spwLHHcRJBbE7kCafpG8ap4Okg8TpIE4HtjuwU4EVlZD3o4EbiEMrIIk94nYWMxgCIP7se7vEEiCPd3x5efn66OjokWIOz66urj7mTByDgWAgGAgGgoFgoA0D8q/n6ull0dvFsRzxO514XBRGMhgIBoKBYCAYCAaMGZDvfcEQJ8bjRPfBQDAQDAQDwUAwcA8Dp/ecW31K3v5MnbD1/UTxRvlvVnfaQQdecUGtF2xecIzN3SsuL7YX+hlbbP/5HnRmuiIWwFvFZ1IFz5zf9K+SOgm94gK9F2xecIwt0isuL7YX+hlbbP/5HnRm6ogLFfAy2F9FfvOkyP9OcdifXyGMV1xQ0h22hXrygmMMvztcja4pL7YX+hlbbP/5zXRm7oi5OBP/va2I2TYnLgpecUFGx9hm6coLjjHojnGtuqa82F7oZ2yx/ee31pm5I5YK2Jpmi/q2f3XMktArLkjwgs0LjrFhesXlxfZCP2OL7T+/qc5MX9ZK3LMi3q2G5ZC/V/5bRe6eLw7YQXvFJbUcecHmBQc6KYNXXGD0gM0DhtLeyrRXbJviMl0Ry8nibHf77skJo9RfFX9IkfxBBa+4UIIXbF5wjC8Mr7i82F7oZ2yx/ed70JmpI5YKuMsgvBHYIa3jn8rjoA/5TWqvuNCVF2xecKCTMnjFBUYP2DxgKO2tTHvFtjmuL35ZK90l/C0t4DRrw3O1e58rK51Xvhcqe1uey3XmHJNMrKRfKl31NbAkQya6HO5BytyUhSn9Xu2eT5QPRalP5FiNS33B789prIfp+JPKb1N68pBkaIqLgVK/rbCBBxnhmDSPJK41xs5GlDcJLXEgoPpDT9W2l+qvun6miGmJK8kYtjdF9MKylvpBhKSjartLbZh3e58bls57tGt6XbXUmfqaNeep/vDLnVMUNxVUAUdQ5eym2qcyjIGVL/08BrDioklY7djizsaFMqqC2pXf9Ny1UTnPqh/q+MuusD7RDJeGfFXKqDQX0TvFez9+UrYpxVb5Glx01RSb+vsomX6jYx3PdeAi+oq8cWiGQ3LPtj21aXH9TFHUDJc6D9ubYnhdWTP9LLE7RFc7izmPrpthU19Lbc/iumqKS9hmz3knsGsRZAw4S+4OWAFhGHxhi+9aM6nNDmrHKhWnuekfUkiGpriE54X6xBByeKUENwiLeMqdLDkaYLuWHL8Xsnyt9NQORFFlfbI1DvUXtrdeLff20FpnGmzvttcaQy92h+JaY1OXXcx7BrgW2d0pJBsFtlMI+Y3pt5+zg3NmYjtX3K1GEyFsk31QxNkxabOC3tTxSoZxaI2Lm5TMzXisfeebYpPuxrsfrNa50bAOTXFYCzuj/9a4wvZmkF9RtbV+KobcW5XW2Hqxvaa4ls55ZitimQe/y8LhspVQhuxYcbRDUB0cL9uxOF62MXHe54o9hqa4wDviCANla2PsxPbBRVNsWWBh4aar1G8+ZXU0wWEl7Ix+m+KSTsL2ZpBfUbWpfirG22eVptg6sr2muLJC5s55lo4Y53pdCIZD5uUmXrTCye7OKc0qiTers5NWdvgISJmnrIdghkv42Y5m1cgz9S2CCTbhYufjqSL/db2PGywTHFsoZDSmGa6wvRHTy7Jm+lkmTtNWZtg2tj0TXHPnPLOtaQnCncadkIS7U5YybA+UbynzzDRvaU/V36zMCpf65Xk6NyS81Ha7BUArbGABkyI3YrwnwE2X2YpffYftfX7Du+qaEl9hew0uOK92l65fk2tqa9uz1Jn6rp7zzBxxrV1LWO5IiOXqF6W/TueOAFTb34x69GnR7yDCHFyqy0TIV8YGY0/5Ix1LTmqhmeJCCMl1pgOxlO+Ozqin8I/iU9XPTjfz/UTluYx6m4QaHKqTZd5ExiWDzsGlumF7S0he0WaOflYMM9UUWza15znYGtveFN5mZTW40mCL5ryTZpIu7EgA7xiH8vmnIkzyrJRNgsZhVTb8rMZigFpcqsdEyE+WeD7+iKg0K8cbxdlB7U1xIZDG+F+dpTpgKJ01Ox20/UNx81CDY3MhFwhQi0v1wvYW8Lu2Sa1+1o4zbq9xu5gbkKu17Y2xts7X6CzVWTTnffGDHq2B3NefADBBs43GCz0AYYLgZa4POpd/g0oZLzJRF2fFc8dPOr9781r5rkIlrn8l9NlYcLU9Hpf1lK/Ehp5+VPyUZOeDHqz8S+ecTm1zqMQRtreNeiZHrdRZ17ZXieHg7A6FVWI7uHmvEtcsu1Ofwwc9unDEk1dbFAYDwUAwEAwEA44ZyI74xDHGgBYMBAPBQDAQDHTPQDji7lUUAgYDwUAwEAx4ZmB4a1rLY75mlQO/9ezmGV4WKo7BQDAQDAQDwcChMyD/yrcUeN+J8EDx4j87By9ukXn/GQAAAABJRU5ErkJggg==",
      "text/latex": [
       "$\\displaystyle \\left[\\begin{matrix}- k_{a1} + k_{a2} + k_{a3} & k_{a1} - k_{a2} + k_{a3} & k_{a1} + k_{a2} - k_{a3}\\end{matrix}\\right]$"
      ],
      "text/plain": [
       "[-k_{a1} + k_{a2} + k_{a3}  k_{a1} - k_{a2} + k_{a3}  k_{a1} + k_{a2} - k_{a3}\n",
       "]"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "tI\n"
     ]
    },
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAT4AAAAZCAYAAACo5kIKAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAFeklEQVR4Ae2db1LVMBTFi8MCUHeAOwBdgboDdQfCDmD8BN8Y3YHuwD87EHeg7ABcAQxL8PxK8qaU+mjfyyXpndyZmKZN05xz09Ob9FU2jo6OvjdNs6MU7fXx8fFFLNS8MlAZqAzMmQHp2YH6v9/BcLgh4fujA7udnXWzMlAZqAy4ZEBatwewRy7RVVCVgcpAZWAJA5tLjq19SOq6pUaYSj9XulL52dqNFtCAF1xecAwNCS/YvODo+yg3LtOIT+CulV4LNGuGp33wcy17weUFx9A48oLNC46+j3LjMhW+DlhenvzslLNvivhXSu18f43OFIdrRSzF4UjkH+goDpsXH62Io39aFv+YCx8DOCAtLeJjGk5ayQrGNQlPwTjW8g8kFIzNi48m4ehXzukfc+ETWKa6THmv+8BnXvaCywuOoeHkBZsXHH0fZcNl+nIjoCTiW0R7EsA3Kr9Q4ol+OGNB9ILLCw4NpzvmBZsXHH0HZcNlGvFJ1BC3xRw+iB7gPyu9C4nyrMwLLi84hgaPF2xecPR9lBuXqfAJLIqOnQpou638h8oI4pzf9HrB5QUHY6xvXrB5wVGUfzb7vYnloMi/VEakxtpbnXfWqdzO4VXGeb91rJ3yhjqTvxYJfSJS3Nf2qPNVj+gyDh5tLuwJWzre/ZQlHjzT/rexMJAnw6XrbKt9+nelxDbLACfa3+VRu0wsGQ56pz4zVkrwD91Jhs2Lj4J/PkCOjLGGvdf+65vNu/+Gc9bVgbsNZ/bPMuGDjFHiMoQq7OOGJrKjnV2R+FlppRta5zFljgI2Wox13pCwNdrPWuO28k/Kp1oyXLrwR6UL9eMLnVDOd4UMtMeUjS0ZDvW7JP9AWzJsasuFj8AhPy3uB20TFPxR+u+HBaqTQgd0iTuW1T9mU10RhjjxVCF6gWy+4OC7YG6QyabziMIQqaz/gYL6kBSX8Jwofe0Q8lTbRH+mlhpHKf6BtNTY1KQLHwnHnrhBcKIh6Dz8V7onYyNT8xL8szm10xPqM+XB4hvd3zfFVgwRsQOlRbQVyCAMP1dCXBAAIsSsQqc+9C0pLuHrR8BEogxIa0uKw7qzE9tPis2RjwhA4n04kdKk1bP7xyziE02ssSBwhMpdi0KGsLWmOggdITdCx5QPsTxQKtFMcAk3D4IuB9bYTXBYd3pk+ybY5u4j7i2l7v2IELLM0n/4jqR55WrZ/WMpfIgZU4TWAuGHKvBiAlFbHNM2EQ5vfqMoqtj+6LlbZl8JZoJL2Il+Xyrx/yE+hOib4CjBQeqDCTZPPhIWprfMLlh/f2jL7h+zqa6IRdVvWRg4t/aFAqFv9y0q6xBxijxUP9s+S1xqmy9ceDiwFsqDwOxJrLZd+oeBYYlNbc/eR8LA2jvBBi8cr+HsIa0E/5gJ31giRQLqT+pGd9yU38OxRrmFc2jTol01u1hgX4qrrdg0f5W/FMYocrFPz7U/7gtVHz4LPliKY47+gckx2ALjbnwkzIgeX0y1D75QbpR3778AO2+mPjHulo690MPJ/rGc6o5iTeC40UmtqRx/FoEjiARNTNchomp/QmJxgTG4Qp0rXb876Ih24eObRb+mtjkGx9Q2x9TXdU39Qx/GYAt1XPhIWBA9fsLCWvoOSdvMMMBXnAXul2pDqDPZP9kjvsA201yeQizuAwJn8EQ6D8Aa5TiNxViEgVfwhOqXyhdvhlUuze7FpQ5T54NwXIbOv1CeZQoSrj+U3Ytjpv4B673YQp3Z+0g4uL+2Qq7sxuQ77qtSzcQ/9W9ulOru2q/KQGUgOQMS+fo3N5KzWhusDFQGZsFA9jW+WbBUO1kZqAy4YqBd41P4x9cS0erf1Y1M1LwyUBmYPQPSN34XG9cxn2j78B810yaM9dNJlwAAAABJRU5ErkJggg==",
      "text/latex": [
       "$\\displaystyle \\left[\\begin{matrix}- k_{a1} + k_{a3} & k_{a1} + k_{a3} & k_{a2} - k_{a3}\\end{matrix}\\right]$"
      ],
      "text/plain": [
       "[-k_{a1} + k_{a3}  k_{a1} + k_{a3}  k_{a2} - k_{a3}]"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "hP\n"
     ]
    },
    {
     "data": {
      "image/png": "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",
      "text/latex": [
       "$\\displaystyle \\left[\\begin{matrix}k_{a1} - k_{a2} & k_{a2} & k_{a3}\\end{matrix}\\right]$"
      ],
      "text/plain": [
       "[k_{a1} - k_{a2}  k_{a2}  k_{a3}]"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "hR_CDML\n"
     ]
    },
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAXUAAAAZCAYAAADOvL5dAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAGFUlEQVR4Ae2d65HUOBSFm6kJgIUM2Ax4RMCSwUIGCxnMFL9m/lFsBmwGu5DBQgZABkAETE0InM8rNZpe2y2rpbZ966pKJUvW455zrGtZds/curi4eLvZbO4rxvDk8vLya8x46gw4A86AM7BMBuSrz2TZi8S681ty6p904kFS6IfOgDPgDDgDK2NAfvw5Jp+szG431xlwBpwBZ2CEgdORcxt5/ts6z/bMQ8Ur5X8dq7+Wc1Zxwb8FbBYwDM0Fy9iGMPeVW+VhCbhGV+oy8FrxiURhj/19nzhrLLOKCy0sYLOAYWheWMY2hLmv3CoPS8A16tQTMXiR+m+St3K4OFy6KH5T7PbGDiR5cdgK8CwOg+tToOJ4E6saz4Zrr1PnIg6amFmpg2fBuNjyIhaHBWPLxrRgDK5PtorjFa1qPDeuvU5dsrD9wjbM9bhEqztrFRdCWMBmAcPQpLCMbQhzX7lVHmbFNfqiNKjASn27Spdz/135R4qsWM5X7Oyt4pIsGwvYLGBAi75gGVsf3qEyqzzMimt0pS6HjePe7g0Fh45AbxSfhUh+VcEqLkSwgM0ChqEJYRnbEOa+cqs8LAHXqFOXGNxxCO9lbHes9J3yOPs1fxFjFRdaWcBmAQNa9AXL2PrwDpVZ5WF2XKdDjIfybm9Ixxj6UQ6924ZR+ln5yb9CVTtuBqzwX+h4b/tQ/4Pq0y43PA32jdWvhivY+DIMdi+kf6j8eswAnedpJ14AadU7ZHQ+/elvPP9Z5U9jZiCtiQ082HilyDHbbq9kA/q3DDUxcO24PhXVkv5weui8rKYx0IJN2b4ltGkxB6vhEqai+bfPqTOhWZHjgB9okDeKRRNa7djGiU4sy0mrDY5xr/NXnamhGi4N/Fp2bh2wjrlQPimO/lArbZMar3LeWdxT+mdaPuG4KjaN+1W2/MX4Svk7E0zmX8g3DFUxyG7Xp6JY4rPGvKymseyZ7FugI70uUnpUfsgcrIZLNr1WnDz/TlIw6bGA4Xi5U7AyY1Lwy1L+TgwETg5qxyoTRzXrHwuTDVVxCc9z9YmQMSAETrmIp9hJSdoA2yvZ8Xdiy10ds2pvFhpgcH2aqVXWcW2N1Z9V31I0/05HZOFRhhC/fPn4X7Zz9JB4prhdTQaheMz9oojjxAGwsp/VicuG3VAbFze8yM3uWMfOV8Um7XafyljBcNNqGapikKGuT0u1yvqurXGZFfVbVcVVOv8GV+rCy94QzptHrTREJ43T7oLq4MTZcsCJ86jOjeBMcYmhKi7w7nCEE+GRadchHoOLqtiiwcLCDTzVN55qkVbFILtdnxYqHdZnVY0PM6Vq6ya4ps6/MaeOo2b53wV1jHM/V+QlJw57e07HrN74QiY6fGW7HyylecqWEJrhEn62XFjNtngPkMNdE2zCxRPZY0X+1n7rm3UTDJDn+sDCIkIzjWdG1wTX1Pk3uP2ijrjr3Aih8xtlIcNjR/pVBnvMcdumr/5sZa1wqV/eP3Bz44Xy9RwAW2EDC5gUuanzXoUbeJMnEfXb5LpTv67PHBdlz5itNO4Z6qhFLXGp7+z5N+jUc9nQYNydiOmqnIn5NpzbYFBuf0upNwWX6uIw+HVt55BCfqM05SQXGlw15Ut2oRcxte+GZsHYb0ofq3504NGuhyqPZaHqcZMcDKrT2avU9TmuPFVGm6JxlQF/dsJ1E6/1n6WVjnJwhaGK5t/JoXaGibMlQPn4eREOgxX8KkMuLtXDYfAZI+8T7gf8rGivSoCrPavg7hPCkvY5bdT/jYs22MzT1VazUAcMqeOnDm3/UZw15GDAQNVzfWZVqnzwXI3LR+hvqXGbzsEcXKFO0fw7eKUeaGHrhZUqL9MwBKfGyu9LMG6jlMnFS0QcA5/8sVXxXen2Cxrllxb24pLBYL4d0q39wgXWJYccbNR5KSzfA5BHSmfbXuohMweD69ND3IqK9mqs69Oqbymaf/4/Sld0dbupzoAz4AwMMaCbm/+P0iFyvNwZcAacgbUycLJWw91uZ8AZcAacgf8z0O2pa9nOr0Bj4Fvk9OVYLPfUGXAGnAFnYEEMyFfzu5H4/u6Ojs9/APiA6HQGGLdGAAAAAElFTkSuQmCC",
      "text/latex": [
       "$\\displaystyle \\left[\\begin{matrix}k_{a1} - k_{a2} + k_{a3} & k_{a2} + k_{a3} & - k_{a1} + k_{a3}\\end{matrix}\\right]$"
      ],
      "text/plain": [
       "[k_{a1} - k_{a2} + k_{a3}  k_{a2} + k_{a3}  -k_{a1} + k_{a3}]"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "hR_spglib\n"
     ]
    },
    {
     "data": {
      "image/png": "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",
      "text/latex": [
       "$\\displaystyle \\left[\\begin{matrix}k_{a1} + k_{a3} & - k_{a1} + k_{a2} + k_{a3} & - k_{a2} + k_{a3}\\end{matrix}\\right]$"
      ],
      "text/plain": [
       "[k_{a1} + k_{a3}  -k_{a1} + k_{a2} + k_{a3}  -k_{a2} + k_{a3}]"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "# Reproduce Table 1.5.4.2 of ITB (2010).\n",
    "ka = Matrix([symbols('k_{a1}'), symbols('k_{a2}'), symbols('k_{a3}')])\n",
    "for name, Q in adjusted.items():\n",
    "    print(name)\n",
    "    Qint = Matrix(np.around(np.array(Q, dtype=float)).astype(int))\n",
    "    display((Qint * ka).T)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "id": "b446e16b",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "mA_oA\n"
     ]
    },
    {
     "data": {
      "image/png": "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",
      "text/latex": [
       "$\\displaystyle \\left[\\begin{matrix}k_{a1}\\\\- 2.0 k_{a3}\\\\2.0 k_{a2}\\end{matrix}\\right]$"
      ],
      "text/plain": [
       "⎡  k_{a1}   ⎤\n",
       "⎢           ⎥\n",
       "⎢-2.0⋅k_{a3}⎥\n",
       "⎢           ⎥\n",
       "⎣2.0⋅k_{a2} ⎦"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "mC_oC\n"
     ]
    },
    {
     "data": {
      "image/png": "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",
      "text/latex": [
       "$\\displaystyle \\left[\\begin{matrix}2.0 k_{a1}\\\\2.0 k_{a2}\\\\k_{a3}\\end{matrix}\\right]$"
      ],
      "text/plain": [
       "⎡2.0⋅k_{a1}⎤\n",
       "⎢          ⎥\n",
       "⎢2.0⋅k_{a2}⎥\n",
       "⎢          ⎥\n",
       "⎣  k_{a3}  ⎦"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "oF_cF\n"
     ]
    },
    {
     "data": {
      "image/png": "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",
      "text/latex": [
       "$\\displaystyle \\left[\\begin{matrix}2.0 k_{a1}\\\\2.0 k_{a2}\\\\2.0 k_{a3}\\end{matrix}\\right]$"
      ],
      "text/plain": [
       "⎡2.0⋅k_{a1}⎤\n",
       "⎢          ⎥\n",
       "⎢2.0⋅k_{a2}⎥\n",
       "⎢          ⎥\n",
       "⎣2.0⋅k_{a3}⎦"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "oI_cI\n"
     ]
    },
    {
     "data": {
      "image/png": "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",
      "text/latex": [
       "$\\displaystyle \\left[\\begin{matrix}2.0 k_{a1}\\\\2.0 k_{a2}\\\\2.0 k_{a3}\\end{matrix}\\right]$"
      ],
      "text/plain": [
       "⎡2.0⋅k_{a1}⎤\n",
       "⎢          ⎥\n",
       "⎢2.0⋅k_{a2}⎥\n",
       "⎢          ⎥\n",
       "⎣2.0⋅k_{a3}⎦"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "tI\n"
     ]
    },
    {
     "data": {
      "image/png": "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",
      "text/latex": [
       "$\\displaystyle \\left[\\begin{matrix}1.0 k_{a1} + 1.0 k_{a2}\\\\- 1.0 k_{a1} + 1.0 k_{a2}\\\\2.0 k_{a3}\\end{matrix}\\right]$"
      ],
      "text/plain": [
       "⎡1.0⋅k_{a1} + 1.0⋅k_{a2}⎤\n",
       "⎢                       ⎥\n",
       "⎢ -k_{a1} + 1.0⋅k_{a2}  ⎥\n",
       "⎢                       ⎥\n",
       "⎣      2.0⋅k_{a3}       ⎦"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "hP\n"
     ]
    },
    {
     "data": {
      "image/png": "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",
      "text/latex": [
       "$\\displaystyle \\left[\\begin{matrix}k_{a1} - k_{a2}\\\\k_{a2}\\\\k_{a3}\\end{matrix}\\right]$"
      ],
      "text/plain": [
       "⎡k_{a1} - k_{a2}⎤\n",
       "⎢               ⎥\n",
       "⎢    k_{a2}     ⎥\n",
       "⎢               ⎥\n",
       "⎣    k_{a3}     ⎦"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "hR_CDML\n"
     ]
    },
    {
     "data": {
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      "text/latex": [
       "$\\displaystyle \\left[\\begin{matrix}2.0 k_{a1} - 1.0 k_{a2}\\\\- 1.0 k_{a1} + 2.0 k_{a2}\\\\3.0 k_{a3}\\end{matrix}\\right]$"
      ],
      "text/plain": [
       "⎡2.0⋅k_{a1} - k_{a2} ⎤\n",
       "⎢                    ⎥\n",
       "⎢-k_{a1} + 2.0⋅k_{a2}⎥\n",
       "⎢                    ⎥\n",
       "⎣     3.0⋅k_{a3}     ⎦"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "hR_spglib\n"
     ]
    },
    {
     "data": {
      "image/png": "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",
      "text/latex": [
       "$\\displaystyle \\left[\\begin{matrix}2.0 k_{a1} - 1.0 k_{a2}\\\\- 1.0 k_{a1} + 2.0 k_{a2}\\\\3.0 k_{a3}\\end{matrix}\\right]$"
      ],
      "text/plain": [
       "⎡2.0⋅k_{a1} - k_{a2} ⎤\n",
       "⎢                    ⎥\n",
       "⎢-k_{a1} + 2.0⋅k_{a2}⎥\n",
       "⎢                    ⎥\n",
       "⎣     3.0⋅k_{a3}     ⎦"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "# Reproduce Table 1.5.4.1 of ITB (2010).\n",
    "# mA_oA result is not consistent with the table (sign of k_2)!\n",
    "for name in transformations.keys():\n",
    "    P = transformations[name]\n",
    "    Q = adjusted[name]\n",
    "    print(name)\n",
    "    display((P.inv().T * Q * ka))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "9791e921",
   "metadata": {},
   "outputs": [],
   "source": []
  }
 ],
 "metadata": {
  "kernelspec": {
   "display_name": "Python 3.9.2 64-bit ('base')",
   "language": "python",
   "name": "python3"
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   "codemirror_mode": {
    "name": "ipython",
    "version": 3
   },
   "file_extension": ".py",
   "mimetype": "text/x-python",
   "name": "python",
   "nbconvert_exporter": "python",
   "pygments_lexer": "ipython3",
   "version": "3.9.2"
  },
  "toc": {
   "base_numbering": 1,
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   "title_cell": "Table of Contents",
   "title_sidebar": "Contents",
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   "toc_position": {},
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   "toc_window_display": false
  },
  "vscode": {
   "interpreter": {
    "hash": "997896bb171f934c03663560f5617afffaf9e983163bfb1351b9671e0b3dfacf"
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 },
 "nbformat": 4,
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}