.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "api/gallery/crystal/defects/plot_02_schottky_defects.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_defects_plot_02_schottky_defects.py>`
        to download the full example code.

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

.. _sphx_glr_api_gallery_crystal_defects_plot_02_schottky_defects.py:


Schottky defects: paired cation and anion vacancies in rock salt
==================================================================

Wagner and Schottky (1930) identified the second point-defect mechanism
of ionic crystals: a cation and an anion leave the bulk together,
keeping the crystal neutral and leaving a vacancy pair behind. The
equilibrium number is :math:`n_S=N\exp(-\Delta H_S/2k_BT)`
(:func:`~chemistrykit.crystal.systems.defects.schottky_defect_concentration`).
In NaCl, the classic Schottky-dominated solid, :math:`\Delta H_S\approx2.3` eV.

.. GENERATED FROM PYTHON SOURCE LINES 14-28

.. code-block:: Python

    import matplotlib.pyplot as plt
    import numpy as np

    from chemistrykit.constants import ELEMENTARY_CHARGE
    from chemistrykit.crystal.systems.defects import schottky_defect_concentration
    from chemistrykit.crystal.visualizers.crystal_plots import plot_defect_concentration_vs_temperature

    N = 2.24e22  # NaCl formula units per cm^3
    T = np.linspace(300.0, 1070.0, 200)  # NaCl melts at 1074 K

    for dh_eV in (2.0, 2.3, 2.6):
        n_melt = schottky_defect_concentration(N, dh_eV * ELEMENTARY_CHARGE, 1070.0)
        print(f"Delta H_S = {dh_eV:.1f} eV: vacancy-pair fraction just below melting = {n_melt / N:.2e}")





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

 .. code-block:: none

    Delta H_S = 2.0 eV: vacancy-pair fraction just below melting = 1.95e-05
    Delta H_S = 2.3 eV: vacancy-pair fraction just below melting = 3.83e-06
    Delta H_S = 2.6 eV: vacancy-pair fraction just below melting = 7.53e-07




.. GENERATED FROM PYTHON SOURCE LINES 29-33

Each Schottky defect is one Na+ vacancy plus one Cl- vacancy, so the
crystal stays stoichiometric -- the ratio of the two vacancy counts is 1
at every temperature -- while the total grows by many orders of
magnitude from room temperature to the melting point:

.. GENERATED FROM PYTHON SOURCE LINES 33-38

.. code-block:: Python

    dh = 2.3 * ELEMENTARY_CHARGE
    n_300 = schottky_defect_concentration(N, dh, 300.0)
    n_1070 = schottky_defect_concentration(N, dh, 1070.0)
    print(f"\nn_S(300 K) = {n_300:.2e} per cm^3,  n_S(1070 K) = {n_1070:.2e} per cm^3,  ratio {n_1070 / n_300:.1e}")





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

 .. code-block:: none


    n_S(300 K) = 1.07e+03 per cm^3,  n_S(1070 K) = 8.58e+16 per cm^3,  ratio 8.0e+13




.. GENERATED FROM PYTHON SOURCE LINES 39-46

.. code-block:: Python

    fig, ax = plt.subplots()
    for dh_eV in (2.0, 2.3, 2.6):
        n = schottky_defect_concentration(N, dh_eV * ELEMENTARY_CHARGE, T)
        plot_defect_concentration_vs_temperature(T, n, ax=ax, label=rf"$\Delta H_S$ = {dh_eV:.1f} eV")
    ax.set_title("Schottky vacancy pairs in NaCl")
    plt.tight_layout()
    plt.show()



.. image-sg:: /api/gallery/crystal/defects/images/sphx_glr_plot_02_schottky_defects_001.png
   :alt: Schottky vacancy pairs in NaCl
   :srcset: /api/gallery/crystal/defects/images/sphx_glr_plot_02_schottky_defects_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_defects_plot_02_schottky_defects.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_schottky_defects.ipynb <plot_02_schottky_defects.ipynb>`

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

      :download:`Download Python source code: plot_02_schottky_defects.py <plot_02_schottky_defects.py>`

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

      :download:`Download zipped: plot_02_schottky_defects.zip <plot_02_schottky_defects.zip>`


.. only:: html

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

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