.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "api/gallery/crystal/defects/plot_01_frenkel_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_01_frenkel_defects.py>`
        to download the full example code.

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

.. _sphx_glr_api_gallery_crystal_defects_plot_01_frenkel_defects.py:


Frenkel defects: vacancy-interstitial pairs in silver chloride
================================================================

Frenkel (1926) argued that at any :math:`T>0` some ions leave their sites
for interstitial gaps, because the configurational entropy gained outweighs
the enthalpy cost at low concentration. The equilibrium number is
:math:`n_F=\sqrt{NN_i}\exp(-\Delta H_F/2k_BT)`
(:func:`~chemistrykit.crystal.systems.defects.frenkel_defect_concentration`).
AgCl, where small Ag+ ions readily occupy interstitial sites, is the
textbook Frenkel-dominated solid (:math:`\Delta H_F\approx1.4` eV).

.. GENERATED FROM PYTHON SOURCE LINES 15-29

.. code-block:: Python

    import matplotlib.pyplot as plt
    import numpy as np

    from chemistrykit.constants import ELEMENTARY_CHARGE, K_B
    from chemistrykit.crystal.systems.defects import frenkel_defect_concentration

    N = 2.3e22  # Ag+ sites per cm^3 in AgCl (approximate)
    delta_h = 1.4 * ELEMENTARY_CHARGE  # J
    T = np.linspace(300.0, 700.0, 200)  # AgCl melts at 728 K

    for T_check in (300.0, 500.0, 700.0):
        nf = frenkel_defect_concentration(N, 2.0 * N, delta_h, T_check)
        print(f"T = {T_check:4.0f} K: fraction of Ag+ displaced to interstitials = {nf / N:.2e}")





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

 .. code-block:: none

    T =  300 K: fraction of Ag+ displaced to interstitials = 2.46e-12
    T =  500 K: fraction of Ag+ displaced to interstitials = 1.24e-07
    T =  700 K: fraction of Ag+ displaced to interstitials = 1.29e-05




.. GENERATED FROM PYTHON SOURCE LINES 30-32

The Arrhenius slope of ln(n_F) vs 1/T is -Delta H_F / (2 k_B): the factor
2 because the vacancy and the interstitial are created together.

.. GENERATED FROM PYTHON SOURCE LINES 32-37

.. code-block:: Python

    n = frenkel_defect_concentration(N, 2.0 * N, delta_h, T)
    slope = np.polyfit(1.0 / T, np.log(n), 1)[0]
    print(f"\nfitted slope {slope:.1f} K,  -Delta H_F / 2k_B = {-delta_h / (2.0 * K_B):.1f} K")
    assert np.isclose(slope, -delta_h / (2.0 * K_B))





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

 .. code-block:: none


    fitted slope -8123.2 K,  -Delta H_F / 2k_B = -8123.2 K




.. GENERATED FROM PYTHON SOURCE LINES 38-51

.. code-block:: Python

    fig, axes = plt.subplots(1, 2, figsize=(11, 4))
    for ratio in (0.5, 2.0, 8.0):
        axes[0].semilogy(T, frenkel_defect_concentration(N, ratio * N, delta_h, T) / N, label=f"N_i = {ratio:g} N")
    axes[0].set_xlabel("T (K)")
    axes[0].set_ylabel(r"$n_F/N$")
    axes[0].set_title(r"Frenkel defects: more interstitial sites, more defects ($\propto\sqrt{N_i}$)")
    axes[0].legend()
    axes[1].plot(1000.0 / T, np.log(n / N))
    axes[1].set_xlabel("1000 / T (1/K)")
    axes[1].set_ylabel(r"$\ln(n_F/N)$")
    axes[1].set_title(r"Arrhenius plot: slope $-\Delta H_F/2k_B$")
    plt.tight_layout()
    plt.show()



.. image-sg:: /api/gallery/crystal/defects/images/sphx_glr_plot_01_frenkel_defects_001.png
   :alt: Frenkel defects: more interstitial sites, more defects ($\propto\sqrt{N_i}$), Arrhenius plot: slope $-\Delta H_F/2k_B$
   :srcset: /api/gallery/crystal/defects/images/sphx_glr_plot_01_frenkel_defects_001.png
   :class: sphx-glr-single-img






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

   **Total running time of the script:** (0 minutes 0.116 seconds)


.. _sphx_glr_download_api_gallery_crystal_defects_plot_01_frenkel_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_01_frenkel_defects.ipynb <plot_01_frenkel_defects.ipynb>`

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

      :download:`Download Python source code: plot_01_frenkel_defects.py <plot_01_frenkel_defects.py>`

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

      :download:`Download zipped: plot_01_frenkel_defects.zip <plot_01_frenkel_defects.zip>`


.. only:: html

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

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