.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "api/gallery/crystal/xrd/plot_03_scherrer_crystallite_size.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_xrd_plot_03_scherrer_crystallite_size.py>`
        to download the full example code.

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

.. _sphx_glr_api_gallery_crystal_xrd_plot_03_scherrer_crystallite_size.py:


Scherrer equation: crystallite size from peak broadening
==========================================================

Scherrer (1918) showed that tiny crystallites give broadened powder
lines, with width :math:`\beta=K\lambda/(\tau\cos\theta)` for
crystallite size :math:`\tau`.
:func:`~chemistrykit.crystal.systems.xrd.scherrer_crystallite_size`
inverts this: here, copper powder patterns are broadened for three
crystallite sizes, and each size is recovered from the measured width of
the (111) line.

.. GENERATED FROM PYTHON SOURCE LINES 15-60



.. image-sg:: /api/gallery/crystal/xrd/images/sphx_glr_plot_03_scherrer_crystallite_size_001.png
   :alt: Cu powder pattern: smaller crystallites, broader lines
   :srcset: /api/gallery/crystal/xrd/images/sphx_glr_plot_03_scherrer_crystallite_size_001.png
   :class: sphx-glr-single-img


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

 .. code-block:: none

    (111) line at 2theta = 43.35 deg
    true size    5.0 nm: FWHM = 1.705 deg -> Scherrer size =    5.0 nm
    true size   20.0 nm: FWHM = 0.425 deg -> Scherrer size =   20.1 nm
    true size  100.0 nm: FWHM = 0.085 deg -> Scherrer size =  100.7 nm






|

.. code-block:: Python

    import matplotlib.pyplot as plt
    import numpy as np

    from chemistrykit.crystal.systems.xrd import powder_xrd_peaks, scherrer_crystallite_size

    wavelength_nm = 0.15418  # Cu-Kalpha
    K = 0.9
    peaks = powder_xrd_peaks("FCC", a=0.3615, wavelength=wavelength_nm, hkl_max=2)
    two_theta = np.linspace(35.0, 80.0, 9001)


    def broadened_pattern(size_nm):
        """Sum of Gaussian lines, each with the Scherrer width for crystallites of `size_nm`."""
        y = np.zeros_like(two_theta)
        for p in peaks:
            fwhm = np.degrees(K * wavelength_nm / (size_nm * np.cos(np.radians(p.two_theta / 2.0))))
            sigma = fwhm / (2.0 * np.sqrt(2.0 * np.log(2.0)))
            y += p.relative_intensity * np.exp(-0.5 * ((two_theta - p.two_theta) / sigma) ** 2)
        return y


    def measured_fwhm(y, center):
        """Full width at half maximum of the line nearest `center`, from half-maximum crossings."""
        window = np.abs(two_theta - center) < 3.0
        x, yy = two_theta[window], y[window]
        above = x[yy >= yy.max() / 2.0]
        return above[-1] - above[0]


    fig, ax = plt.subplots(figsize=(8, 4))
    first = peaks[0]
    print(f"(111) line at 2theta = {first.two_theta:.2f} deg")
    for size in (5.0, 20.0, 100.0):
        y = broadened_pattern(size)
        ax.plot(two_theta, y / y.max(), label=f"{size:.0f} nm crystallites")
        fwhm = measured_fwhm(y, first.two_theta)
        tau = scherrer_crystallite_size(fwhm, first.two_theta, wavelength_nm, shape_factor=K)
        print(f"true size {size:6.1f} nm: FWHM = {fwhm:.3f} deg -> Scherrer size = {tau:6.1f} nm")
        assert abs(tau - size) / size < 0.05
    ax.set_xlabel(r"$2\theta$ (degrees)")
    ax.set_ylabel("normalized intensity")
    ax.set_title("Cu powder pattern: smaller crystallites, broader lines")
    ax.legend()
    plt.tight_layout()
    plt.show()


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

   **Total running time of the script:** (0 minutes 0.052 seconds)


.. _sphx_glr_download_api_gallery_crystal_xrd_plot_03_scherrer_crystallite_size.py:

.. only:: html

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

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

      :download:`Download Jupyter notebook: plot_03_scherrer_crystallite_size.ipynb <plot_03_scherrer_crystallite_size.ipynb>`

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

      :download:`Download Python source code: plot_03_scherrer_crystallite_size.py <plot_03_scherrer_crystallite_size.py>`

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

      :download:`Download zipped: plot_03_scherrer_crystallite_size.zip <plot_03_scherrer_crystallite_size.zip>`


.. only:: html

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

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