.. 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_02_goldschmidt_tolerance_factor.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_02_goldschmidt_tolerance_factor.py>`
        to download the full example code.

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

.. _sphx_glr_api_gallery_crystal_crystal_chemistry_plot_02_goldschmidt_tolerance_factor.py:


Goldschmidt's tolerance factor: which perovskites are cubic
=============================================================

Goldschmidt (1926) noted that in an ideal cubic :math:`ABX_3` perovskite
the A-X distance equals :math:`\sqrt2` times the B-X distance, so hard
spheres fit exactly when
:math:`t=(r_A+r_X)/[\sqrt2(r_B+r_X)]=1`
(:func:`~chemistrykit.crystal.systems.crystal_chemistry.goldschmidt_tolerance_factor`).
Titanates with A-cations of different sizes span the range from tilted
to cubic to ferroelectrically distorted.

.. GENERATED FROM PYTHON SOURCE LINES 15-36

.. code-block:: Python

    import matplotlib.pyplot as plt
    import numpy as np

    from chemistrykit.crystal.systems.crystal_chemistry import goldschmidt_tolerance_factor

    r_O = 140.0
    perovskites = {
        # name: (r_A (CN 12), r_B (CN 6), observed room-temperature structure), Shannon radii in pm
        "CaTiO3": (134.0, 60.5, "orthorhombic (tilted)"),
        "SrTiO3": (144.0, 60.5, "cubic"),
        "BaTiO3": (161.0, 60.5, "tetragonal (ferroelectric)"),
        "CaZrO3": (134.0, 72.0, "orthorhombic (tilted)"),
        "KNbO3": (164.0, 64.0, "orthorhombic (ferroelectric)"),
    }
    ts = {}
    for name, (r_a, r_b, observed) in perovskites.items():
        ts[name] = goldschmidt_tolerance_factor(r_a, r_b, r_O)
        print(f"{name:7s} t = {ts[name]:.3f}   observed: {observed}")
    assert abs(ts["SrTiO3"] - 1.0) < 0.01
    assert ts["CaTiO3"] < ts["SrTiO3"] < ts["BaTiO3"]





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

 .. code-block:: none

    CaTiO3  t = 0.966   observed: orthorhombic (tilted)
    SrTiO3  t = 1.002   observed: cubic
    BaTiO3  t = 1.062   observed: tetragonal (ferroelectric)
    CaZrO3  t = 0.914   observed: orthorhombic (tilted)
    KNbO3   t = 1.054   observed: orthorhombic (ferroelectric)




.. GENERATED FROM PYTHON SOURCE LINES 37-51

.. code-block:: Python

    r_a = np.linspace(100.0, 180.0, 200)
    fig, ax = plt.subplots(figsize=(8, 4))
    ax.plot(r_a, goldschmidt_tolerance_factor(r_a, 60.5, r_O), "k-", label=r"$ATiO_3$ (B = Ti$^{4+}$)")
    ax.axhspan(0.9, 1.0, color="C2", alpha=0.15, label="roughly cubic")
    ax.axhline(1.0, color="C2", lw=1)
    for name, (r_a_i, _r_b, _) in perovskites.items():
        ax.plot(r_a_i, ts[name], "o")
        ax.annotate(name, (r_a_i, ts[name]), textcoords="offset points", xytext=(5, 5))
    ax.set_xlabel(r"A-cation radius $r_A$ (pm)")
    ax.set_ylabel("tolerance factor t")
    ax.set_title("Goldschmidt tolerance factor of perovskites")
    ax.legend(loc="lower right")
    plt.tight_layout()
    plt.show()



.. image-sg:: /api/gallery/crystal/crystal_chemistry/images/sphx_glr_plot_02_goldschmidt_tolerance_factor_001.png
   :alt: Goldschmidt tolerance factor of perovskites
   :srcset: /api/gallery/crystal/crystal_chemistry/images/sphx_glr_plot_02_goldschmidt_tolerance_factor_001.png
   :class: sphx-glr-single-img






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

   **Total running time of the script:** (0 minutes 0.062 seconds)


.. _sphx_glr_download_api_gallery_crystal_crystal_chemistry_plot_02_goldschmidt_tolerance_factor.py:

.. only:: html

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

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

      :download:`Download Jupyter notebook: plot_02_goldschmidt_tolerance_factor.ipynb <plot_02_goldschmidt_tolerance_factor.ipynb>`

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

      :download:`Download Python source code: plot_02_goldschmidt_tolerance_factor.py <plot_02_goldschmidt_tolerance_factor.py>`

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

      :download:`Download zipped: plot_02_goldschmidt_tolerance_factor.zip <plot_02_goldschmidt_tolerance_factor.zip>`


.. only:: html

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

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