.. 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_03_hauy_rational_indices.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_03_hauy_rational_indices.py>`
        to download the full example code.

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

.. _sphx_glr_api_gallery_crystal_crystal_systems_plot_03_hauy_rational_indices.py:


Hauy's law of rational indices: Miller indices from face intercepts
====================================================================

Hauy (1784-1801) argued that every natural crystal face cuts the three
axes at intercepts in small whole-number ratios.
:func:`~chemistrykit.crystal.systems.crystal_systems.miller_indices_from_intercepts`
turns those rational intercepts into the integer Miller indices
:math:`(hkl)` by taking reciprocals and clearing fractions; the
:math:`(hkl)` triple then fixes each face's interplanar spacing.

.. GENERATED FROM PYTHON SOURCE LINES 14-35

.. code-block:: Python

    import math

    import matplotlib.pyplot as plt
    import numpy as np

    from chemistrykit.crystal.systems.crystal_systems import miller_indices_from_intercepts
    from chemistrykit.crystal.systems.xrd import d_spacing_cubic

    faces = {
        "cube face": (1, math.inf, math.inf),
        "dodecahedron face": (1, 1, math.inf),
        "octahedron face": (1, 1, 1),
        "intercepts 1, 2, 3": (1, 2, 3),
        "intercepts 1/2, 1, inf": (0.5, 1, math.inf),
        "intercepts 2, 3, 6": (2, 3, 6),
    }
    a = 5.64  # NaCl, Angstrom
    for name, intercepts in faces.items():
        hkl = miller_indices_from_intercepts(*intercepts)
        print(f"{name:24s} intercepts={str(intercepts):22s} -> (hkl)={hkl}   d={d_spacing_cubic(a, *hkl):.3f} A")





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

 .. code-block:: none

    cube face                intercepts=(1, inf, inf)          -> (hkl)=(1, 0, 0)   d=5.640 A
    dodecahedron face        intercepts=(1, 1, inf)            -> (hkl)=(1, 1, 0)   d=3.988 A
    octahedron face          intercepts=(1, 1, 1)              -> (hkl)=(1, 1, 1)   d=3.256 A
    intercepts 1, 2, 3       intercepts=(1, 2, 3)              -> (hkl)=(6, 3, 2)   d=0.806 A
    intercepts 1/2, 1, inf   intercepts=(0.5, 1, inf)          -> (hkl)=(2, 1, 0)   d=2.522 A
    intercepts 2, 3, 6       intercepts=(2, 3, 6)              -> (hkl)=(3, 2, 1)   d=1.507 A




.. GENERATED FROM PYTHON SOURCE LINES 36-38

The intercept ratios are rational, so the reciprocals clear to small
integers: intercepts (2, 3, 6) give reciprocals (1/2, 1/3, 1/6) = (3, 2, 1)/6.

.. GENERATED FROM PYTHON SOURCE LINES 38-40

.. code-block:: Python

    assert miller_indices_from_intercepts(2, 3, 6) == (3, 2, 1)








.. GENERATED FROM PYTHON SOURCE LINES 41-42

Draw a few faces as planes cutting the unit cube at their intercepts.

.. GENERATED FROM PYTHON SOURCE LINES 42-67

.. code-block:: Python

    fig = plt.figure(figsize=(9, 3.5))
    for i, (hkl, color) in enumerate((((1, 0, 0), "C0"), ((1, 1, 0), "C1"), ((1, 1, 1), "C2"))):
        ax = fig.add_subplot(1, 3, i + 1, projection="3d")
        for s, e in [((0, 0, 0), (1, 0, 0)), ((0, 0, 0), (0, 1, 0)), ((0, 0, 0), (0, 0, 1))]:
            ax.plot(*zip(s, e, strict=True), "k-", lw=0.8)
        h, k, l = hkl
        g = np.linspace(0, 1, 2)
        if l != 0:
            # the (111) face is the triangle through the three unit intercepts
            ax.plot_trisurf(np.array([1, 0, 0]), np.array([0, 1, 0]), np.array([0, 0, 1]), color=color, alpha=0.5)
        elif k != 0:
            X, Z = np.meshgrid(g, g)
            ax.plot_surface(X, (1 - h * X) / k, Z, color=color, alpha=0.5)
        else:
            Y, Z = np.meshgrid(g, g)
            ax.plot_surface(np.ones_like(Y) / h, Y, Z, color=color, alpha=0.5)
        ax.set_title(f"({h}{k}{l})")
        ax.set_xlim(0, 1)
        ax.set_ylim(0, 1)
        ax.set_zlim(0, 1)
        ax.set_xlabel("a")
        ax.set_ylabel("b")
        ax.set_zlabel("c")
    plt.tight_layout()
    plt.show()



.. image-sg:: /api/gallery/crystal/crystal_systems/images/sphx_glr_plot_03_hauy_rational_indices_001.png
   :alt: (100), (110), (111)
   :srcset: /api/gallery/crystal/crystal_systems/images/sphx_glr_plot_03_hauy_rational_indices_001.png
   :class: sphx-glr-single-img






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

   **Total running time of the script:** (0 minutes 0.087 seconds)


.. _sphx_glr_download_api_gallery_crystal_crystal_systems_plot_03_hauy_rational_indices.py:

.. only:: html

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

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      :download:`Download Python source code: plot_03_hauy_rational_indices.py <plot_03_hauy_rational_indices.py>`

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

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