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      "source": [
        "\n# Bulk Structures and Relaxations\n\nHere, we create bulk structures and optimize them to their ideal bulk properties\n"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "metadata": {
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      "source": [
        "import matplotlib.pyplot as plt\nimport numpy as np\n\nfrom ase.build import bulk\nfrom ase.calculators.emt import EMT\nfrom ase.eos import EquationOfState\nfrom ase.filters import FrechetCellFilter\nfrom ase.optimize import BFGS\nfrom ase.units import kJ\nfrom ase.visualize.plot import plot_atoms"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {},
      "source": [
        "## Setting up bulk structures\n\nASE provides three frameworks for setting up bulk structures:\n\n * :func:`ase.build.bulk`.  Knows lattice types\n   and lattice constants for elemental bulk structures\n   and a few compounds, but\n   with limited customization.\n\n * :func:`ase.spacegroup.crystal`.  Creates atoms\n   from typical crystallographic information such as spacegroup,\n   lattice parameters, and basis.\n\n * :mod:`ase.lattice`.  Creates atoms explicitly from lattice and basis.\n\nLet's run a simple bulk calculation. We use :func:`ase.build.bulk`\nto get a primitive cell of silver, and then visualize it. Silver is known\nto form an FCC structure, so presumably the function returned a primitive\nFCC cell. You can, e.g., use the ASE GUI to repeat the structure and\nrecognize the A-B-C stacking.\n\n"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "metadata": {
        "collapsed": false
      },
      "outputs": [],
      "source": [
        "atoms = bulk('Ag')\n\nfig, ax = plt.subplots()\nplot_atoms(atoms * (3, 3, 3), ax=ax)\nax.set_xlabel(r'$x (\\AA)$')\nax.set_ylabel(r'$y (\\AA)$')\nfig.tight_layout()\n\n# For interactive use of the ASE GUI:\n# view(atoms * (3,3,3))"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {},
      "source": [
        "ASE should also be able to verify that it really is a primitive FCC cell\nand tell us what lattice constant was chosen:\n\n"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "metadata": {
        "collapsed": false
      },
      "outputs": [],
      "source": [
        "print(f'Bravais lattice: {atoms.cell.get_bravais_lattice()}')"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {},
      "source": [
        "## Periodicity\n\nPeriodic structures in ASE are represented using ``atoms.cell``\nand ``atoms.pbc``.\n* The cell is a :class:`~ase.cell.Cell` object which represents\nthe crystal lattice with three vectors.\n* ``pbc``  is an array of three booleans indicating whether the system\nis periodic in each direction.\n\n"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "metadata": {
        "collapsed": false
      },
      "outputs": [],
      "source": [
        "print('Cell:\\n', atoms.cell.round(3))\nprint('Periodicity: ', atoms.pbc)"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {},
      "source": [
        "## Equation of state\n\nWe can find the optimal lattice parameter and calculate the bulk modulus\nby doing an equation-of-state calculation.  This means sampling the energy\nand lattice constant over a range of values to get the minimum as well\nas the curvature, which gives us the bulk modulus.\n\nThe online ASE docs already provide a tutorial on how to perform\nequation-of-state calculations: (`eos_example`)\n\nFirst, we calculate the volume and potential energy, whilst scaling the\natoms' cell. Here, we use ase's empirical calculator ``EMT``:\n\n\n"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "metadata": {
        "collapsed": false
      },
      "outputs": [],
      "source": [
        "calc = EMT()\ncell = atoms.get_cell()\n\nvolumes = []\nenergies = []\nfor x in np.linspace(0.95, 1.05, 5):\n    atoms_copy = atoms.copy()\n    atoms_copy.calc = calc\n    atoms_copy.set_cell(cell * x, scale_atoms=True)\n    atoms_copy.get_potential_energy()\n    volumes.append(atoms_copy.get_volume())\n    energies.append(atoms_copy.get_potential_energy())"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {},
      "source": [
        "Then, via :func:`ase.eos.EquationOfState`, we can calculate and plot\nthe bulk modulus:\n\n\n"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "metadata": {
        "collapsed": false
      },
      "outputs": [],
      "source": [
        "eos = EquationOfState(volumes, energies)\nv0, e0, B = eos.fit()\n\nax = eos.plot()\nax.axhline(e0, linestyle='--', alpha=0.5, color='black')\nax.axvline(v0, linestyle='--', alpha=0.5, color='red')\n\nprint(f'Minimum Volume = {v0:.3f}AA^3')\nprint(f'Minimum Energy = {e0:.3f}eV')\nprint(f'Bulk modulus   = {B / kJ * 1.0e24:.3f} GPa')\nplt.show()"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {},
      "source": [
        "## Bulk Optimization\n\nWe can also find the optimal, relaxed cell via variable-cell relaxation.\nThis requires setting a filter, in this case the\n:func:`ase.filters.FrechetCellFilter` filter, which allows minimizing both the\natomic forces and the unit cell stress.\n\n\n"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "metadata": {
        "collapsed": false
      },
      "outputs": [],
      "source": [
        "original_lattice = atoms.cell.get_bravais_lattice()\ncalc = EMT()\natoms.calc = calc\n\nopt = BFGS(FrechetCellFilter(atoms), trajectory='opt.Ag.traj')\nopt.run(fmax=0.05)\nprint('\\n')\nprint(f'Original_Lattice:   {original_lattice}')\nprint(f'Final Lattice:      {atoms.cell.get_bravais_lattice()}')\nprint(f'Final Cell Volume:  {atoms.get_volume():.3f}AA^3')"
      ]
    }
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