.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "api/gallery/crystal/crystal_systems/plot_04_bravais_cubic_lattices.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_crystal_systems_plot_04_bravais_cubic_lattices.py>`
        to download the full example code.

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

.. _sphx_glr_api_gallery_crystal_crystal_systems_plot_04_bravais_cubic_lattices.py:


Bravais lattices: the 14 lattices and the three cubic centerings
==================================================================

Bravais (1848) showed there are exactly 14 distinct three-dimensional
lattices: the 7 crystal systems combined with the centerings each allows
(:data:`~chemistrykit.crystal.systems.crystal_systems.BRAVAIS_LATTICES`).
The cubic system has three -- primitive (P), body-centered (I), and
face-centered (F) -- drawn here with
:func:`~chemistrykit.crystal.systems.crystal_systems.cubic_lattice_points`.

.. GENERATED FROM PYTHON SOURCE LINES 14-25

.. code-block:: Python

    import matplotlib.pyplot as plt
    import numpy as np

    from chemistrykit.crystal.systems.crystal_systems import BRAVAIS_LATTICES, cubic_lattice_points

    for system, centerings in BRAVAIS_LATTICES.items():
        print(f"{system:13s} {', '.join(centerings)}")
    total = sum(len(c) for c in BRAVAIS_LATTICES.values())
    print(f"\nTotal Bravais lattices: {total}")
    assert total == 14





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

 .. code-block:: none

    cubic         P, I, F
    tetragonal    P, I
    orthorhombic  P, C, I, F
    hexagonal     P
    trigonal      R
    monoclinic    P, C
    triclinic     P

    Total Bravais lattices: 14




.. GENERATED FROM PYTHON SOURCE LINES 26-28

Lattice points per conventional cell (corners shared by 8 cells, faces by
2) and the nearest-neighbor distance, measured from the generated points:

.. GENERATED FROM PYTHON SOURCE LINES 28-36

.. code-block:: Python

    for centering, name in (("P", "primitive"), ("I", "body-centered"), ("F", "face-centered")):
        pts = cubic_lattice_points(centering, a=1.0, n_cells=4)
        per_cell = np.sum(np.all(pts < 4.0 - 1e-9, axis=1)) / 4**3
        dists = np.linalg.norm(pts[:, None, :] - pts[None, :, :], axis=-1)
        d_nn = dists[dists > 1e-9].min()
        n_nn = np.sum(np.isclose(dists[np.argmin(np.linalg.norm(pts - 2.0, axis=1))], d_nn))
        print(f"c{centering} ({name:13s}): {per_cell:.0f} lattice point(s)/cell, nearest neighbor {d_nn:.4f} a, {n_nn} neighbors")





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

 .. code-block:: none

    cP (primitive    ): 1 lattice point(s)/cell, nearest neighbor 1.0000 a, 6 neighbors
    cI (body-centered): 2 lattice point(s)/cell, nearest neighbor 0.8660 a, 8 neighbors
    cF (face-centered): 4 lattice point(s)/cell, nearest neighbor 0.7071 a, 12 neighbors




.. GENERATED FROM PYTHON SOURCE LINES 37-49

.. code-block:: Python

    fig = plt.figure(figsize=(10, 3.6))
    edges = [(p, q) for p in np.ndindex(2, 2, 2) for q in np.ndindex(2, 2, 2) if np.sum(np.abs(np.subtract(p, q))) == 1 and p < q]
    for i, (centering, title) in enumerate((("P", "cP (simple cubic)"), ("I", "cI (body-centered)"), ("F", "cF (face-centered)"))):
        ax = fig.add_subplot(1, 3, i + 1, projection="3d")
        for p, q in edges:
            ax.plot(*zip(p, q, strict=True), color="gray", lw=0.8)
        pts = cubic_lattice_points(centering)
        ax.scatter(pts[:, 0], pts[:, 1], pts[:, 2], s=60, color="C0", depthshade=False)
        ax.set_title(title)
        ax.set_axis_off()
    plt.tight_layout()
    plt.show()



.. image-sg:: /api/gallery/crystal/crystal_systems/images/sphx_glr_plot_04_bravais_cubic_lattices_001.png
   :alt: cP (simple cubic), cI (body-centered), cF (face-centered)
   :srcset: /api/gallery/crystal/crystal_systems/images/sphx_glr_plot_04_bravais_cubic_lattices_001.png
   :class: sphx-glr-single-img






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

   **Total running time of the script:** (0 minutes 0.056 seconds)


.. _sphx_glr_download_api_gallery_crystal_crystal_systems_plot_04_bravais_cubic_lattices.py:

.. only:: html

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

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

      :download:`Download Jupyter notebook: plot_04_bravais_cubic_lattices.ipynb <plot_04_bravais_cubic_lattices.ipynb>`

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

      :download:`Download Python source code: plot_04_bravais_cubic_lattices.py <plot_04_bravais_cubic_lattices.py>`

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

      :download:`Download zipped: plot_04_bravais_cubic_lattices.zip <plot_04_bravais_cubic_lattices.zip>`


.. only:: html

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

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