SW_curves_1_single.ipynb

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   "cell_type": "markdown",
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   "source": [
    "# **Calculating Stoner-Wohlfarth Curves**"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The Stoner-Wohlfarth model is given in the paper \"A Mechanism of Magnetic Hysteresis in Heterogeneous Alloys\":\n",
    "\n",
    "> [E.C. Stoner and E.P. Wohlfarth, Phil. Trans. Royal Soc. London 240 (1948) 599](https://doi.org/10.1098/rsta.1948.0007)\n",
    "\n",
    "<br>\n",
    "\n",
    "- The SW-model considers magnetisation processes governed by **<font color=red>rotations</font>**  of the magnetisation vector.\n",
    "- Magnetisation processes based on domains and domain wall movement are **NOT** considered.\n",
    "\n",
    "The model is therefore strictly valid for:\n",
    "- single-domain particles ($d < d_{c}$) or \n",
    "- ensembles of non-interacting, single-domain particles\n",
    "\n",
    "But we can also apply the SW-model in cases where *domain wall motion is not relevant*:\n",
    "- hard axis magnetisation curve of a single crystal with uniaxial anisotropy\n",
    "- magnetisation curve of a polycrystal at applied field strengths above those where domain wall movement occurs (neglecting particle-particle interactions).\n",
    "\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<br>\n",
    "\n",
    "Consider a single-domain particle with uniaxial magnetocrystalline anisotropy.\n",
    "\n",
    "- In zero applied field, $H$, the magnetisation, $M_{0}$, will be parallel to the easy axis.\n",
    "\n",
    "- If we apply a field, $H$, perpendicular to the easy axis, $M_{0}$ will be forced to rotate away from the easy axis, towards $H$."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<img src=\"resources/SW-angles-schematic.png\" width=\"400\" align=\"left\"/>"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<br>\n",
    "\n",
    "The angle between the field and the magnetisation, $\\theta_{M}$, results from miminising the sum of the **anisotropy energy** and the **Zeeman (field) energy**:\n",
    "\n",
    "## $E_{v} = K_{1}sin^{2}(\\theta_{M}-\\theta_{H})  + K_{2}sin^{4}(\\theta_{M}-\\theta_{H}) - M_{0}Hcos{\\theta_{M}}$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "where:\n",
    "\n",
    "$E_{v}$ = the total energy per unit volume\n",
    "\n",
    "$M_{0}$ = the spontaneous magnetisation of the material\n",
    "\n",
    "$K_{1}$, $K_{2}$ = second and fourth order magnetocrystalline anisotropy constants\n",
    "\n",
    "$\\theta_{H}$ = angle between the applied field and the easy axis\n",
    "\n",
    "$\\theta_{M}$ = angle between the applied field and the magnetisation\n",
    "\n",
    "$H$ = applied field strength"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<br>\n",
    "\n",
    "Before we can write down the equation in code, we need to import some things we will need:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {},
   "outputs": [],
   "source": [
    "import numpy\n",
    "import matplotlib.pyplot as plt\n",
    "from matplotlib.ticker import StrMethodFormatter\n",
    "from scipy.optimize import minimize\n",
    "import ipywidgets as widgets\n",
    "from ipywidgets import interactive, fixed\n",
    "\n",
    "%matplotlib inline\n",
    "%config InlineBackend.figure_format = 'svg'\n",
    "\n",
    "plt.rc('font', size=10)\n",
    "plt.rc('figure', figsize=(6,4))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {},
   "outputs": [],
   "source": [
    "def get_energy_terms(thetaM, thetaH, H, M0, K1, K2):\n",
    "    \"\"\"\n",
    "    Calculate the anisotropy energy, the Zeeman energy and the total energy, given\n",
    "    the arguments.\n",
    "    \n",
    "    Args:\n",
    "        thetaM = angle in degrees between the magnetisation and the applied field\n",
    "        thetaH = angle in degrees between the easy axis and the applied field\n",
    "        H = applied field in Oe\n",
    "        M0 = spontaneous magnetisation in erg/Oe.cm^3 (== kA/m)\n",
    "        K1 = second order anisotropy constant in erg/cm^3\n",
    "        K2 = fourth order anisotropy constant in erg/cm^3\n",
    "    \n",
    "    Returns: \n",
    "        AE = float, the anisotropy energy per unit volume for the conditions given, unit: erg/cm^3.\n",
    "        ZE = float, the Zeeman energy per unit volume for the conditions given, unit: erg/cm^3.\n",
    "        TE = float, the total energy per unit volume for the conditions given, unit: erg/cm^3.\n",
    "    \"\"\"\n",
    "    t1 = K1*numpy.sin(numpy.deg2rad(thetaM-thetaH))**2\n",
    "    t2 = K2*numpy.sin(numpy.deg2rad(thetaM-thetaH))**4\n",
    "\n",
    "    AE = t1+t2\n",
    "    ZE = M0*H*numpy.cos(numpy.deg2rad(thetaM))\n",
    "    TE = AE-ZE\n",
    "    \n",
    "    return AE, ZE, TE"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<br>\n",
    "\n",
    "### Example with $Y_{2}Fe_{14}B$\n",
    "This material has a tetragonal crystal structure and uniaxial magnetocrystalline anisotropy, where the <001> axis is the easy axis.\n",
    "\n",
    "Here are values for for the constants for $Y_{2}Fe_{14}B$ extracted from [Givord *et al.* Solid State Comm. 51 (1984) 857-860](https://doi.org/10.1016/0038-1098(84)91087-1)."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {},
   "outputs": [],
   "source": [
    "M0 = 1053 # erg/Oecm^3 == kA/m (~ 1.4 T )\n",
    "K1 = 1.25e7 # erg/cm^3 \n",
    "K2 = 0.05e7 # erg/cm^3 "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Let's examine how the energy per unit volume, $E_{v}$, changes as a function of the angle between the field and the magnetisation, $\\theta_{M}$ for different applied field strengths, $H$, using an interactive plot."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "metadata": {},
   "outputs": [
    {
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       "model_id": "76e410eba9f44448813ec90c231397a5",
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      },
      "text/plain": [
       "interactive(children=(Dropdown(description='thetaH', options=(90, 0), value=90), IntSlider(value=0, descriptio…"
      ]
     },
     "execution_count": 20,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "def plot_func(thetaH, H, M0, K1, K2):\n",
    "    thetaM = numpy.arange(91)\n",
    "    AE, ZE, TE = get_energy_terms(thetaM, thetaH, H, M0, K1, K2)\n",
    "    \n",
    "    fig, ax = plt.subplots()\n",
    "    \n",
    "    ax.plot(thetaM, AE, 'b-', lw=2.0, label='Anisotropy Energy')\n",
    "    ax.plot(thetaM, ZE, 'r-', lw=2.0, label='Zeeman Energy')\n",
    "    ax.plot(thetaM, TE, 'g-', lw=2.0, label='Total Energy')\n",
    "\n",
    "    ax.grid(which=\"major\")\n",
    "    ax.set_xlabel(r\"$\\theta_{M} \\ /\\ \\degree$\")\n",
    "    ax.set_ylabel(r\"$E_{v}\\ /\\ erg \\cdot cm^{-3}$\")\n",
    "    ax.legend(loc=\"upper right\")\n",
    "    ax.set_xlim(xmin=0.0, xmax=90.0)\n",
    "    ax.yaxis.set_major_formatter(StrMethodFormatter('{x: ,.2e}'))\n",
    "\n",
    "\n",
    "    \n",
    "interactive_plot = interactive(plot_func, thetaH=[90,0], \n",
    "                               H=widgets.IntSlider(value=0.0, min=0.0, max=28000, \n",
    "                                                   step=1000, description=\"H / Oe\"),\n",
    "                               M0=fixed(M0), K1=fixed(K1), K2=fixed(K2))\n",
    "\n",
    "output = interactive_plot.children[-1]\n",
    "output.layout.height = '360px'\n",
    "interactive_plot"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Let's redefine our fuction so that it only returns the total energy (we don't need the other terms individually now):"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "metadata": {},
   "outputs": [],
   "source": [
    "def SW_E(thetaM, thetaH, H, M0, K1, K2):\n",
    "    \"\"\"\n",
    "    Calculate the energy per unit volume resulting from the\n",
    "    balance of the anisotropy energy and Zeeman energy, given\n",
    "    the arguments.\n",
    "    \n",
    "    Args:\n",
    "        thetaM = angle in degrees between the magnetisation and the applied field\n",
    "        thetaH = angle in degrees between the easy axis and the applied field\n",
    "        H = applied field in Oe\n",
    "        M0 = spontaneous magnetisation in erg/Oe.cm^3 (== kA/m)\n",
    "        K1 = second order anisotropy constant in erg/cm^3\n",
    "        K2 = fourth order anisotropy constant in erg/cm^3\n",
    "    \n",
    "    Returns: \n",
    "        E = float, the energy per unit volume for the conditions given, unit: erg/cm^3.\n",
    "    \"\"\"\n",
    "    t1 = K1*numpy.sin(numpy.deg2rad(thetaM-thetaH))**2\n",
    "    t2 = K2*numpy.sin(numpy.deg2rad(thetaM-thetaH))**4\n",
    "    t3 = M0*H*numpy.cos(numpy.deg2rad(thetaM))\n",
    "    return t1+t2-t3"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We can use `scipy.optimize.minimize` to find the minimum of $E_{v}$ for a particular applied field and angle $\\theta_{H}$ like this:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "metadata": {},
   "outputs": [],
   "source": [
    "q = minimize(SW_E, 90, args=(90, 8000, M0, K1, K2), method=\"Nelder-Mead\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 23,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "       message: Optimization terminated successfully.\n",
       "       success: True\n",
       "        status: 0\n",
       "           fun: -1412943.6346844418\n",
       "             x: [ 7.049e+01]\n",
       "           nit: 20\n",
       "          nfev: 40\n",
       " final_simplex: (array([[ 7.049e+01],\n",
       "                       [ 7.049e+01]]), array([-1.413e+06, -1.413e+06]))"
      ]
     },
     "execution_count": 23,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "q"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 24,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "70.48951721191406"
      ]
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     "execution_count": 24,
     "metadata": {},
     "output_type": "execute_result"
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   ],
   "source": [
    "q.x[0]"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<br>\n",
    "\n",
    "We can use `scipy.optimize.minimize` in a **loop** to find the minimum of $E_{v}$ for a several applied field values like this:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 25,
   "metadata": {},
   "outputs": [],
   "source": [
    "def getData_single(FR, M0, K1, K2, thetaH):\n",
    "    \"\"\"\n",
    "    For each value of the field in the field range (FR), find the minimum\n",
    "    of the absolute value of the derivative of the energy equation with\n",
    "    respect to the angle theta.\n",
    "    \n",
    "    Args:\n",
    "        FR : array of applied field values in Oe\n",
    "        thetaH = angle in degrees between the easy axis and the applied field\n",
    "        M0 = spontaneous magnetisation in erg/Oe.cm^3 (== kA/m)\n",
    "        K1 = second order anisotropy constant in erg/cm^3\n",
    "        K2 = fourth order anisotropy constant in erg/cm^3\n",
    "        \n",
    "    Returns:\n",
    "        D : a list of angles thetaM in degrees at which the derivative of the energy\n",
    "            equation is zero for the field values defined in FR\n",
    "    \"\"\"\n",
    "    D = []\n",
    "    IG = thetaH\n",
    "    for H in FR:\n",
    "        q = minimize(SW_E, IG, args=(thetaH, H, M0, K1, K2), method=\"Nelder-Mead\")\n",
    "        D.append(q.x[0])\n",
    "        IG = q.x[0]\n",
    "\n",
    "        \n",
    "    return D"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 26,
   "metadata": {},
   "outputs": [],
   "source": [
    "# field range (start, stop, step) in Oe\n",
    "FR = numpy.arange(0, 6.6e4, 1.0e3)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 27,
   "metadata": {},
   "outputs": [],
   "source": [
    "D = getData_single(FR, M0, K1, K2, 90.0)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Now,  $\\cos\\theta_{M} = \\frac{M}{M_{0}}$, so to plot M(H) we need to multiply the cosine of the angles, $\\theta_{M}$ in D by $M_{0}$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 28,
   "metadata": {},
   "outputs": [
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   "source": [
    "plt.plot(FR, numpy.cos(numpy.deg2rad(D))*M0, 'r-')\n",
    "plt.xlabel(\"H / Oe\")\n",
    "plt.ylabel(\"M / kA/m\")\n",
    "plt.xlim(xmin=0)\n",
    "plt.ylim(ymin=0)\n",
    "plt.axvline(x=25800, ls=':')\n",
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  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Experimental results for a $Y_{2}Fe_{14}B$  single crystal were measured by [Givord *et al.* Solid State Comm. 51 (1984) 857-860](https://doi.org/10.1016/0038-1098(84)91087-1).\n",
    "\n",
    "Two text files containing experimental data points extracted from the above paper are in the folder \"resources\" and can be loaded using  `numpy.loadtxt`."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 29,
   "metadata": {},
   "outputs": [],
   "source": [
    "# load Givord's measurements of a Y2Fe14B SX at 300K\n",
    "\n",
    "Ge = numpy.loadtxt(\"resources/Givord_Y2Fe14B_001_conv.txt\")\n",
    "Gh = numpy.loadtxt(\"resources/Givord_Y2Fe14B_100_and_110_conv.txt\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 30,
   "metadata": {},
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      "text/plain": [
       "<Figure size 600x400 with 1 Axes>"
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     "metadata": {},
     "output_type": "display_data"
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   ],
   "source": [
    "plt.plot(Gh[:,0],Gh[:,1],'ko', label=\"Experiment, [100] or [110]\")\n",
    "plt.plot(FR, numpy.cos(numpy.deg2rad(D))*M0, 'r-', label=\"SW curve, hard axis\")\n",
    "plt.xlabel(\"H / Oe\")\n",
    "plt.ylabel(\"M / kA/m\")\n",
    "plt.xlim(xmin=0)\n",
    "plt.ylim(ymin=0)\n",
    "_ = plt.legend()"
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  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We have very good agreement between the experimentally measured data for a single crystal and our calculated Stoner-Wohlfarth curve."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<br>\n",
    "\n",
    "Now we can use the same code to calculate an easy axis curve to compare with the experimental data:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 31,
   "metadata": {},
   "outputs": [],
   "source": [
    "D1 = getData_single(FR, M0, K1, K2, 0.0)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 32,
   "metadata": {},
   "outputs": [
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   ],
   "source": [
    "plt.plot(Gh[:,0],Gh[:,1],'ko', label=\"Experiment, [100] or [110]\")\n",
    "plt.plot(FR, numpy.cos(numpy.deg2rad(D))*M0, 'r-', label=\"SW curve, hard axis\")\n",
    "plt.plot(Ge[:,0],Ge[:,1],'go', label=\"Experiment, [001]\")\n",
    "plt.plot(FR, numpy.cos(numpy.deg2rad(D1))*M0, 'b-', label=\"SW curve, easy axis\")\n",
    "\n",
    "plt.xlabel(\"H / Oe\")\n",
    "plt.ylabel(\"M / kA/m\")\n",
    "plt.xlim(xmin=0)\n",
    "plt.ylim(ymin=0)\n",
    "_ = plt.legend()"
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  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The discrepancy between the SW-curve for the easy axis and the experimental data at fields below ~ 5000 Oe is due to the formation of domains in the sample. Domains are not accounted for in the Stoner-Wohlfarth model."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  }
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