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.. _sphx_glr_api_gallery_crystal_xrd_plot_02_laue_interference.py:


Laue's interference: sharp diffraction spots from a periodic array
====================================================================

Von Laue (1912) predicted that a crystal, as a periodic array of
scatterers, should diffract X-rays like a grating, and Friedrich and
Knipping saw the discrete spots. Summing the waves from :math:`N`
periodically spaced scatterers with
:func:`~chemistrykit.crystal.systems.xrd.structure_factor` gives the Laue
interference function: intensity :math:`N^2` at integer :math:`h`
(the Laue condition) and almost nothing in between, ever sharper as
:math:`N` grows.

.. GENERATED FROM PYTHON SOURCE LINES 16-36

.. code-block:: Python

    import matplotlib.pyplot as plt
    import numpy as np

    from chemistrykit.crystal.systems.xrd import structure_factor

    h = np.linspace(-0.5, 2.5, 1201)
    fig, axes = plt.subplots(1, 2, figsize=(11, 4))
    for N in (3, 6, 12):
        row = [(n, 0.0, 0.0) for n in range(N)]
        intensity = np.array([abs(structure_factor((hh, 0, 0), row)) ** 2 for hh in h])
        axes[0].plot(h, intensity / N**2, label=f"N = {N}")
        peak = abs(structure_factor((1, 0, 0), row)) ** 2
        between = abs(structure_factor((0.5, 0, 0), row)) ** 2
        print(f"N={N:3d}: I(h=1) = {peak:7.1f} (= N^2 = {N * N}),  I(h=1/2) = {between:.3f}")
        assert np.isclose(peak, N * N)
    axes[0].set_xlabel("h (scattering vector, reciprocal-lattice units)")
    axes[0].set_ylabel(r"$|F|^2 / N^2$")
    axes[0].set_title("1D Laue interference function")
    axes[0].legend()




.. image-sg:: /api/gallery/crystal/xrd/images/sphx_glr_plot_02_laue_interference_001.png
   :alt: 1D Laue interference function
   :srcset: /api/gallery/crystal/xrd/images/sphx_glr_plot_02_laue_interference_001.png
   :class: sphx-glr-single-img


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

 .. code-block:: none

    N=  3: I(h=1) =     9.0 (= N^2 = 9),  I(h=1/2) = 1.000
    N=  6: I(h=1) =    36.0 (= N^2 = 36),  I(h=1/2) = 0.000
    N= 12: I(h=1) =   144.0 (= N^2 = 144),  I(h=1/2) = 0.000

    <matplotlib.legend.Legend object at 0x1197dc1a0>



.. GENERATED FROM PYTHON SOURCE LINES 37-39

In two dimensions the same interference gives a grid of sharp spots --
the kind of pattern Friedrich and Knipping recorded on their plate:

.. GENERATED FROM PYTHON SOURCE LINES 39-49

.. code-block:: Python

    N = 8
    grid = [(i, j, 0.0) for i in range(N) for j in range(N)]
    hk = np.linspace(-2.2, 2.2, 221)
    pattern = np.array([[abs(structure_factor((hh, kk, 0), grid)) ** 2 for hh in hk] for kk in hk])
    axes[1].imshow(np.log10(pattern + 1.0), extent=(hk[0], hk[-1], hk[0], hk[-1]), origin="lower", cmap="magma")
    axes[1].set_xlabel("h")
    axes[1].set_ylabel("k")
    axes[1].set_title(f"2D array of {N}x{N} scatterers: Laue spots")
    plt.tight_layout()
    plt.show()



.. image-sg:: /api/gallery/crystal/xrd/images/sphx_glr_plot_02_laue_interference_002.png
   :alt: plot 02 laue interference
   :srcset: /api/gallery/crystal/xrd/images/sphx_glr_plot_02_laue_interference_002.png
   :class: sphx-glr-single-img






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   **Total running time of the script:** (0 minutes 0.645 seconds)


.. _sphx_glr_download_api_gallery_crystal_xrd_plot_02_laue_interference.py:

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      :download:`Download Python source code: plot_02_laue_interference.py <plot_02_laue_interference.py>`

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