docs/uniaxialanisotropy.ipynb

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   "source": [
    "# Tutorial 03: Uniaxial anisotropy energy term\n",
    "\n",
    "Uniaxial anisotropy energy density is computed as\n",
    "\n",
    "$$w_\\text{ua} = -K (\\mathbf{m}\\cdot\\mathbf{u})^{2}$$\n",
    "\n",
    "where $\\mathbf{m}$ is the normalised ($|\\mathbf{m}|=1$) magnetisation, $K$ is the uniaxial anisotropy constant, and $\\mathbf{u}$ is the anisotropy axis. Uniaxial anisotropy energy term tends to align all magnetic moments parallel or antiparallel to the anisotropy axis.\n",
    "\n",
    "In `oommfc`, $\\mathbf{m}$ is a part of the magnetisation field `system.m`. Therefore, only uniaxial anisotropy constant $K$ and axis $\\mathbf{u}$ should be provided as input parameters to uniquely define the energy term. Both $K$ and $\\mathbf{u}$ can be constant in space or spatially varying.\n",
    "\n",
    "## Spatially constant $K$ and $\\mathbf{u}$\n",
    "\n",
    "Let us start by assembling a simple simple simulation where neither $K$ nor $\\mathbf{u}$ vary in space. The sample is a \"one-dimensional\" chain of magnetic moments."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "import discretisedfield as df\n",
    "import micromagneticmodel as mm\n",
    "\n",
    "import oommfc as oc\n",
    "\n",
    "p1 = (-10e-9, 0, 0)\n",
    "p2 = (10e-9, 1e-9, 1e-9)\n",
    "cell = (1e-9, 1e-9, 1e-9)\n",
    "region = df.Region(p1=p1, p2=p2)\n",
    "mesh = df.Mesh(region=region, cell=cell)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The mesh is"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
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     },
     "metadata": {},
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   ],
   "source": [
    "# NBVAL_IGNORE_OUTPUT\n",
    "mesh.k3d()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The system has a Hamiltonian, which consists of only uniaxial anisotropy energy term."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [],
   "source": [
    "K = 1e5  # uniaxial anisotropy constant (J/m**3)\n",
    "u = (0, 0, 1)  # uniaxial anisotropy axis\n",
    "system = mm.System(name=\"uniaxialanisotropy_constant_K_u\")\n",
    "system.energy = mm.UniaxialAnisotropy(K=K, u=u)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We are going to minimise the system's energy using `oommfc.MinDriver` later. Therefore, we do not have to define the system's dynamics equation. Finally, we need to define the system's magnetisation (`system.m`). We are going to make it random with $M_\\text{s}=8\\times10^{5} \\,\\text{Am}^{-1}$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [],
   "source": [
    "import random\n",
    "\n",
    "import discretisedfield as df\n",
    "\n",
    "Ms = 8e5  # saturation magnetisation (A/m)\n",
    "\n",
    "\n",
    "def m_fun(pos):\n",
    "    return [2 * random.random() - 1 for i in range(3)]\n",
    "\n",
    "\n",
    "system.m = df.Field(mesh, dim=3, value=m_fun, norm=Ms)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The magnetisation, we set is"
   ]
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   "execution_count": 5,
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       "Output()"
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     "metadata": {},
     "output_type": "display_data"
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   ],
   "source": [
    "# NBVAL_IGNORE_OUTPUT\n",
    "system.m.k3d.vector(color_field=system.m.z)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Now, we can minimise the system's energy by using `oommfc.MinDriver`."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Running OOMMF (ExeOOMMFRunner) [2021/09/22 13:52]...  <1> mmarchive killed\n",
      " <2> mmarchive killed\n",
      "(1.1 s)\n"
     ]
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   ],
   "source": [
    "# NBVAL_IGNORE_OUTPUT\n",
    "md = oc.MinDriver()\n",
    "md.drive(system)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We expect that now all magnetic moments are aligned parallel or antiparallel to the anisotropy axis (in the $z$-direction)."
   ]
  },
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   "cell_type": "code",
   "execution_count": 7,
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   ],
   "source": [
    "# NBVAL_IGNORE_OUTPUT\n",
    "system.m.k3d.vector(color_field=system.m.z)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Spatially varying $\\mathbf{K}$\n",
    "\n",
    "There are two different ways how a parameter can be made spatially varying, by using:\n",
    "1. Dictionary\n",
    "2. `discretisedfield.Field`\n",
    "\n",
    "### Dictionary\n",
    "\n",
    "In order to define a parameter using a dictionary, regions must be defined in the mesh. Regions are defined as a dictionary, whose keys are the strings and values are `discretisedfield.Region` objects, which take two corner points of the region as input parameters. "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
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   "source": [
    "p1 = (-10e-9, 0, 0)\n",
    "p2 = (10e-9, 1e-9, 1e-9)\n",
    "cell = (1e-9, 1e-9, 1e-9)\n",
    "subregions = {\n",
    "    \"region1\": df.Region(p1=(-10e-9, 0, 0), p2=(0, 1e-9, 1e-9)),\n",
    "    \"region2\": df.Region(p1=(0, 0, 0), p2=(10e-9, 1e-9, 1e-9)),\n",
    "}\n",
    "region = df.Region(p1=p1, p2=p2)\n",
    "mesh = df.Mesh(region=region, cell=cell, subregions=subregions)"
   ]
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  {
   "cell_type": "code",
   "execution_count": 9,
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       "Output()"
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   ],
   "source": [
    "# NBVAL_IGNORE_OUTPUT\n",
    "mesh.k3d.subregions()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Let us say that there is no uniaxial anisotropy ($K=0$) in region 1, whereas in region 2 it is $K=10^{5} \\,\\text{Jm}^{-3}$. $\\mathbf{u}$ is in the $z$-direction. `K` is now defined as a dictionary:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [],
   "source": [
    "K = {\"region1\": 0, \"region2\": 1e5}"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The system object is"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [],
   "source": [
    "system = mm.System(name=\"uniaxialanisotropy_dict_K\")\n",
    "system.energy = mm.UniaxialAnisotropy(K=K, u=u)\n",
    "system.m = df.Field(mesh, dim=3, value=m_fun, norm=Ms)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Its magnetisation is"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
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   ],
   "source": [
    "# NBVAL_IGNORE_OUTPUT\n",
    "system.m.k3d.vector(color_field=system.m.z)"
   ]
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   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "After we minimise the energy"
   ]
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   "cell_type": "code",
   "execution_count": 13,
   "metadata": {},
   "outputs": [
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     "name": "stdout",
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     "text": [
      "Running OOMMF (ExeOOMMFRunner) [2021/09/22 13:52]...  <1> mmarchive killed\n",
      " <2> mmarchive killed\n",
      "(1.3 s)\n"
     ]
    }
   ],
   "source": [
    "# NBVAL_IGNORE_OUTPUT\n",
    "md.drive(system)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The magnetisation is as we expected."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
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   "source": [
    "# NBVAL_IGNORE_OUTPUT\n",
    "system.m.k3d.vector(color_field=system.m.z)"
   ]
  },
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   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### `discretisedfield.Field`\n",
    "\n",
    "Let us define the spatailly varying uniaxial anisotropy, so that\n",
    "\n",
    "$\\mathbf{u}(x, y, z) = \\left\\{\n",
    "\\begin{array}{ll}\n",
    "(0, 0, 1) & x \\le 0 \\\\\n",
    "(1, 0, 0) & x > 0 \\\\\n",
    "\\end{array}\n",
    "\\right. $\n",
    "\n",
    "The value of `u` for the spatially varying anisotropy is set using a Python function."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {},
   "outputs": [],
   "source": [
    "def u_fun(pos):\n",
    "    x, y, z = pos\n",
    "    if x <= 0:\n",
    "        return (0, 0, 1)\n",
    "    else:\n",
    "        return (1, 0, 0)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The uniaxial anisotropy parameters are"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {},
   "outputs": [],
   "source": [
    "K = 1e5\n",
    "u = df.Field(mesh, dim=3, value=u_fun)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The system is"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {},
   "outputs": [],
   "source": [
    "system = mm.System(name=\"uniaxialanisotropy_field_u\")\n",
    "system.energy = mm.UniaxialAnisotropy(K=K, u=u)\n",
    "system.m = df.Field(mesh, dim=3, value=m_fun, norm=Ms)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "and its magnetisation is"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
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   ],
   "source": [
    "# NBVAL_IGNORE_OUTPUT\n",
    "system.m.k3d.vector(color_field=system.m.z)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "After the energy minimisation, the magnetisation is:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Running OOMMF (ExeOOMMFRunner) [2021/09/22 13:52]...  <1> mmarchive killed\n",
      " <2> mmarchive killed\n",
      "(1.4 s)\n"
     ]
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   ],
   "source": [
    "# NBVAL_IGNORE_OUTPUT\n",
    "md.drive(system)\n",
    "system.m.k3d.vector(color_field=system.m.z)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Higher-order uniaxial anisotropy\n",
    "\n",
    "In order to define higher-order anisotropy term:\n",
    "\n",
    "$$w_\\text{ua} = -K_{1} (\\mathbf{m}\\cdot\\mathbf{u})^{2} - K_{2} (\\mathbf{m}\\cdot\\mathbf{u})^{4}$$\n",
    "\n",
    "anisotropy constants `K1` and `K2` must be passed. Similar to previous example, all parameters (`K1`, `K2`, and `u`) can be spatially varying (dictionary or field)."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "metadata": {},
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     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Running OOMMF (ExeOOMMFRunner) [2021/09/22 13:53]...  <1> mmarchive killed\n",
      " <2> mmarchive killed\n",
      "(1.5 s)\n"
     ]
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   "source": [
    "# NBVAL_IGNORE_OUTPUT\n",
    "K1 = 1e5\n",
    "K2 = 1e3\n",
    "u = df.Field(mesh, dim=3, value=u_fun)\n",
    "\n",
    "system = mm.System(name=\"uniaxialanisotropy_higher_order\")\n",
    "system.energy = mm.UniaxialAnisotropy(K1=K1, K2=K2, u=u)\n",
    "system.m = df.Field(mesh, dim=3, value=m_fun, norm=Ms)\n",
    "\n",
    "md.drive(system)\n",
    "system.m.k3d.vector(color_field=system.m.z)"
   ]
  }
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