.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "api/gallery/crystal/crystal_chemistry/plot_01_pauling_radius_ratio.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_chemistry_plot_01_pauling_radius_ratio.py>`
        to download the full example code.

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

.. _sphx_glr_api_gallery_crystal_crystal_chemistry_plot_01_pauling_radius_ratio.py:


Pauling's radius-ratio rule: predicting coordination numbers
==============================================================

Pauling's first rule (1929): each cation is surrounded by a polyhedron of
anions whose size is fixed by the radius ratio :math:`r_+/r_-`. A
polyhedron is stable only while the anions still touch the cation, which
sets the limits :math:`\sqrt2-1=0.414` (octahedral), :math:`\sqrt3-1=0.732`
(cubic), and so on.
:func:`~chemistrykit.crystal.systems.crystal_chemistry.radius_ratio_coordination`
applies the rule; here it is tested against observed structures.

.. GENERATED FROM PYTHON SOURCE LINES 15-37

.. code-block:: Python

    import matplotlib.pyplot as plt

    from chemistrykit.crystal.systems.crystal_chemistry import RADIUS_RATIO_LIMITS, radius_ratio_coordination

    compounds = {
        # name: (r_cation, r_anion in pm, observed cation coordination)
        "BeO (wurtzite)": (27.0, 138.0, 4),  # Be2+ CN4, O2- CN4
        "ZnS (zinc blende)": (60.0, 184.0, 4),  # Zn2+ CN4, S2- CN6
        "NaCl (rock salt)": (102.0, 181.0, 6),
        "MgO (rock salt)": (72.0, 140.0, 6),
        "TiO2 (rutile)": (60.5, 140.0, 6),
        "CsCl": (174.0, 181.0, 8),  # Cs+ CN8
        "CaF2 (fluorite)": (112.0, 131.0, 8),  # Ca2+ CN8, F- CN4
    }
    print(f"{'compound':20s} {'r+/r-':>6s}  {'predicted':>20s}  observed CN")
    hits = 0
    for name, (rc, ra, observed) in compounds.items():
        p = radius_ratio_coordination(rc, ra)
        hits += p.coordination_number == observed
        print(f"{name:20s} {p.radius_ratio:6.3f}  {p.geometry:>14s} (CN {p.coordination_number})  {observed}")
    print(f"\n{hits}/{len(compounds)} coordination numbers predicted correctly")





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

 .. code-block:: none

    compound              r+/r-             predicted  observed CN
    BeO (wurtzite)        0.196  trigonal planar (CN 3)  4
    ZnS (zinc blende)     0.326     tetrahedral (CN 4)  4
    NaCl (rock salt)      0.564      octahedral (CN 6)  6
    MgO (rock salt)       0.514      octahedral (CN 6)  6
    TiO2 (rutile)         0.432      octahedral (CN 6)  6
    CsCl                  0.961           cubic (CN 8)  8
    CaF2 (fluorite)       0.855           cubic (CN 8)  8

    6/7 coordination numbers predicted correctly




.. GENERATED FROM PYTHON SOURCE LINES 38-41

The rule is a guide, not a law: BeO above falls just below the
tetrahedral limit, and LiI (r+/r- = 0.35) is rock salt, not tetrahedral,
because real ions are not hard spheres.

.. GENERATED FROM PYTHON SOURCE LINES 41-43

.. code-block:: Python

    print(f"LiI: predicted CN {radius_ratio_coordination(76.0, 220.0).coordination_number}, observed CN 6")





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

 .. code-block:: none

    LiI: predicted CN 4, observed CN 6




.. GENERATED FROM PYTHON SOURCE LINES 44-59

.. code-block:: Python

    fig, ax = plt.subplots(figsize=(9, 3.5))
    edges = [lim[0] for lim in RADIUS_RATIO_LIMITS] + [1.0]
    for (lo, cn, geom), hi in zip(RADIUS_RATIO_LIMITS, edges[1:], strict=True):
        ax.axvspan(lo, hi, alpha=0.15, color=f"C{cn % 10}")
        ax.text((lo + hi) / 2.0, 9.3, f"{geom}\nCN {cn}", ha="center", va="top", fontsize=8)
    for name, (rc, ra, observed) in compounds.items():
        ax.plot(rc / ra, observed, "ko")
        ax.annotate(name.split(" ")[0], (rc / ra, observed), textcoords="offset points", xytext=(4, -12), fontsize=8)
    ax.set_xlim(0, 1.0)
    ax.set_ylim(1, 9.5)
    ax.set_xlabel(r"radius ratio $r_+/r_-$")
    ax.set_ylabel("observed cation CN")
    ax.set_title("Pauling's radius-ratio bands vs. observed structures")
    plt.tight_layout()
    plt.show()



.. image-sg:: /api/gallery/crystal/crystal_chemistry/images/sphx_glr_plot_01_pauling_radius_ratio_001.png
   :alt: Pauling's radius-ratio bands vs. observed structures
   :srcset: /api/gallery/crystal/crystal_chemistry/images/sphx_glr_plot_01_pauling_radius_ratio_001.png
   :class: sphx-glr-single-img






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

   **Total running time of the script:** (0 minutes 0.047 seconds)


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.. only:: html

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

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

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