.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "api/gallery/crystal/packing/plot_01_packing_fractions.py"
.. LINE NUMBERS ARE GIVEN BELOW.

.. only:: html

    .. note::
        :class: sphx-glr-download-link-note

        :ref:`Go to the end <sphx_glr_download_api_gallery_crystal_packing_plot_01_packing_fractions.py>`
        to download the full example code.

.. rst-class:: sphx-glr-example-title

.. _sphx_glr_api_gallery_crystal_packing_plot_01_packing_fractions.py:


Kepler's conjecture: packing efficiency of SC, BCC, FCC, and HCP
==================================================================

Kepler (1611) guessed that no stacking of equal spheres fills space more
densely than the layered (face-centered cubic / hexagonal close)
packing, with fraction :math:`\pi/\sqrt{18}\approx0.7405`. Comparing the
four textbook lattices shows the close-packed pair at exactly that bound.

The four lattices implemented here
(:mod:`chemistrykit.crystal.systems.packing`) share the
:class:`~chemistrykit.crystal.core.base_system.LatticePacking` interface:
each derives its atomic packing factor from pure geometry (touching
spheres, atoms-per-cell, and a lattice-constant-to-radius relation).
FCC and ideal HCP -- both close-packed, differing only in stacking
sequence -- turn out to share the same maximum packing fraction.

.. GENERATED FROM PYTHON SOURCE LINES 20-42

.. code-block:: Python

    import matplotlib.pyplot as plt
    import numpy as np

    from chemistrykit.crystal.systems.packing import (
        BodyCenteredCubicPacking,
        FaceCenteredCubicPacking,
        HexagonalClosePacking,
        SimpleCubicPacking,
    )
    from chemistrykit.crystal.visualizers.crystal_plots import plot_packing_fractions

    lattices = {
        "SC": SimpleCubicPacking(),
        "BCC": BodyCenteredCubicPacking(),
        "FCC": FaceCenteredCubicPacking(),
        "HCP (ideal)": HexagonalClosePacking(),
    }

    for name, lattice in lattices.items():
        pf = lattice.packing_fraction()
        print(f"{name:12s} packing fraction = {pf:.4f}   coordination = {lattice.coordination_number:2d}   atoms/cell = {lattice.atoms_per_cell}")





.. rst-class:: sphx-glr-script-out

 .. code-block:: none

    SC           packing fraction = 0.5236   coordination =  6   atoms/cell = 1.0
    BCC          packing fraction = 0.6802   coordination =  8   atoms/cell = 2.0
    FCC          packing fraction = 0.7405   coordination = 12   atoms/cell = 4.0
    HCP (ideal)  packing fraction = 0.7405   coordination = 12   atoms/cell = 2.0




.. GENERATED FROM PYTHON SOURCE LINES 43-46

FCC and ideal HCP are both close-packed and share the same packing
fraction, even though the underlying models (cubic vs. hexagonal cell,
4 vs. 2 atoms per cell) look nothing alike:

.. GENERATED FROM PYTHON SOURCE LINES 46-54

.. code-block:: Python

    fcc_pf = lattices["FCC"].packing_fraction()
    hcp_pf = lattices["HCP (ideal)"].packing_fraction()
    print(f"\nFCC packing fraction:  {fcc_pf:.9f}")
    print(f"HCP packing fraction:  {hcp_pf:.9f}")
    print(f"Match: {abs(fcc_pf - hcp_pf) < 1e-9}")
    print(f"Kepler's bound pi/sqrt(18) = {np.pi / np.sqrt(18.0):.9f}")
    assert abs(fcc_pf - np.pi / np.sqrt(18.0)) < 1e-12





.. rst-class:: sphx-glr-script-out

 .. code-block:: none


    FCC packing fraction:  0.740480490
    HCP packing fraction:  0.740480490
    Match: True
    Kepler's bound pi/sqrt(18) = 0.740480490




.. GENERATED FROM PYTHON SOURCE LINES 55-57

A real HCP metal's actual c/a ratio deviates from the ideal value,
which lowers its packing fraction below the FCC/ideal-HCP maximum:

.. GENERATED FROM PYTHON SOURCE LINES 57-60

.. code-block:: Python

    zinc_hcp = HexagonalClosePacking(c_over_a=1.856)  # zinc's anomalous c/a
    print(f"\nZinc-like HCP (c/a=1.856) packing fraction: {zinc_hcp.packing_fraction():.4f} (below the ideal {hcp_pf:.4f})")





.. rst-class:: sphx-glr-script-out

 .. code-block:: none


    Zinc-like HCP (c/a=1.856) packing fraction: 0.6515 (below the ideal 0.7405)




.. GENERATED FROM PYTHON SOURCE LINES 61-68

.. code-block:: Python

    names = list(lattices.keys())
    fractions = [lattices[name].packing_fraction() for name in names]
    ax = plot_packing_fractions(names, fractions)
    ax.axhline(np.pi / np.sqrt(18.0), color="k", ls="--", lw=1, label=r"Kepler bound $\pi/\sqrt{18}$")
    ax.legend()
    plt.tight_layout()
    plt.show()



.. image-sg:: /api/gallery/crystal/packing/images/sphx_glr_plot_01_packing_fractions_001.png
   :alt: Atomic packing factor by lattice type
   :srcset: /api/gallery/crystal/packing/images/sphx_glr_plot_01_packing_fractions_001.png
   :class: sphx-glr-single-img






.. rst-class:: sphx-glr-timing

   **Total running time of the script:** (0 minutes 0.031 seconds)


.. _sphx_glr_download_api_gallery_crystal_packing_plot_01_packing_fractions.py:

.. only:: html

  .. container:: sphx-glr-footer sphx-glr-footer-example

    .. container:: sphx-glr-download sphx-glr-download-jupyter

      :download:`Download Jupyter notebook: plot_01_packing_fractions.ipynb <plot_01_packing_fractions.ipynb>`

    .. container:: sphx-glr-download sphx-glr-download-python

      :download:`Download Python source code: plot_01_packing_fractions.py <plot_01_packing_fractions.py>`

    .. container:: sphx-glr-download sphx-glr-download-zip

      :download:`Download zipped: plot_01_packing_fractions.zip <plot_01_packing_fractions.zip>`


.. only:: html

 .. rst-class:: sphx-glr-signature

    `Gallery generated by Sphinx-Gallery <https://sphinx-gallery.github.io>`_