.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "sl_examples/plot_09_PakePattern.py"
.. LINE NUMBERS ARE GIVEN BELOW.

.. only:: html

    .. note::
        :class: sphx-glr-download-link-note

        :ref:`Go to the end <sphx_glr_download_sl_examples_plot_09_PakePattern.py>`
        to download the full example code.

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

.. _sphx_glr_sl_examples_plot_09_PakePattern.py:


.. _PakePattern_Example:

Pake Pattern Simulation (#9)
=============================

.. GENERATED FROM PYTHON SOURCE LINES 9-14

This example demonstrates how to simulate a Pake pattern for a dipolar-coupled
electron spin pair using the SpinLab theory module.
The Pake pattern is the powder-averaged dipolar spectrum observed in DEER/PELDOR
experiments. The two characteristic features are the horns at :math:`\pm\nu_{dd}`
and the shoulders at :math:`\pm 2\nu_{dd}`.

.. GENERATED FROM PYTHON SOURCE LINES 14-26

.. code-block:: Python


    import spinlab as sl
    import numpy as np
    from spinlab.plotting.colors import (
        BrukerPacific,
        BrukerOcean,
        BrukerIce,
        BrukerOrange,
        BrukerDolomite,
        BrukerGranite,
    )








.. GENERATED FROM PYTHON SOURCE LINES 27-33

Define the inter-spin distance and calculate the dipolar coupling frequency.
The function uses the DEER convention:

.. math::

    \nu_{dd} = \frac{\mu_0}{4\pi} \frac{g_1 g_2 \mu_B^2}{h r^3}

.. GENERATED FROM PYTHON SOURCE LINES 33-38

.. code-block:: Python


    r = 2e-9  # inter-spin distance in meters (2 nm)
    nu_dd = sl.distance_to_dipolar_coupling(r, unit="MHz")
    print(f"Dipolar coupling: {nu_dd:.3f} MHz")





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

 .. code-block:: none

    Dipolar coupling: 6.505 MHz




.. GENERATED FROM PYTHON SOURCE LINES 39-41

Set up the frequency axis and the powder-averaging orientations using
Gauss-Legendre quadrature on the unit sphere.

.. GENERATED FROM PYTHON SOURCE LINES 41-45

.. code-block:: Python


    freq = np.linspace(-40e6, 40e6, 4096)  # frequency axis in Hz — wide enough for all distances
    theta, phi, weights = sl.sphere_quadrature(n_theta=500, n_phi=1)








.. GENERATED FROM PYTHON SOURCE LINES 46-48

Simulate the Pake pattern. A Lorentzian linewidth of 0.8 MHz is applied
via an exponential decay in the time domain before Fourier transformation.

.. GENERATED FROM PYTHON SOURCE LINES 48-52

.. code-block:: Python


    linewidth = 0.8e6  # Hz
    spectrum = sl.pake_pattern(freq, theta, nu_dd * 1e6, linewidth, weights)








.. GENERATED FROM PYTHON SOURCE LINES 53-55

Plot the result. The horns of the Pake pattern appear at
:math:`\pm\nu_{dd}` and the shoulders at :math:`\pm 2\nu_{dd}`.

.. GENERATED FROM PYTHON SOURCE LINES 55-72

.. code-block:: Python


    freq_MHz = freq / 1e6
    spectrum = spectrum - spectrum.min()  # baseline to zero

    sl.plt.figure()
    sl.plt.plot(freq_MHz, spectrum, color=BrukerPacific)
    sl.plt.axvline(nu_dd, color=BrukerOrange, linestyle="--", linewidth=0.8, label=r"$\pm\nu_{dd}$")
    sl.plt.axvline(-nu_dd, color=BrukerOrange, linestyle="--", linewidth=0.8)
    sl.plt.axvline(2 * nu_dd, color=BrukerDolomite, linestyle=":", linewidth=0.8, label=r"$\pm 2\nu_{dd}$")
    sl.plt.axvline(-2 * nu_dd, color=BrukerDolomite, linestyle=":", linewidth=0.8)
    sl.plt.xlabel("Frequency (MHz)")
    sl.plt.ylabel("Intensity (arb. u.)")
    sl.plt.title(f"Pake Pattern, r = {r*1e9:.0f} nm, $\\nu_{{dd}}$ = {nu_dd:.2f} MHz")
    sl.plt.legend()
    sl.plt.grid(ls=":")
    sl.plt.show()




.. image-sg:: /sl_examples/images/sphx_glr_plot_09_PakePattern_001.png
   :alt: Pake Pattern, r = 2 nm, $\nu_{dd}$ = 6.51 MHz
   :srcset: /sl_examples/images/sphx_glr_plot_09_PakePattern_001.png
   :class: sphx-glr-single-img





.. GENERATED FROM PYTHON SOURCE LINES 73-78

Distance dependence
-------------------

The dipolar coupling scales as :math:`r^{-3}`. The following plot shows
Pake patterns for a range of inter-spin distances.

.. GENERATED FROM PYTHON SOURCE LINES 78-96

.. code-block:: Python


    distances_nm = [1.5, 2.0, 2.5, 3.0]
    colors = [BrukerPacific, BrukerOcean, BrukerIce, BrukerOrange]

    sl.plt.figure()
    for r_nm, color in zip(distances_nm, colors):
        nu = sl.distance_to_dipolar_coupling(r_nm * 1e-9, unit="MHz")
        spec = sl.pake_pattern(freq, theta, nu * 1e6, linewidth, weights)
        spec = spec - spec.min()  # baseline to zero
        spec = spec / spec.max()  # normalize peak to 1
        sl.plt.plot(freq_MHz, spec, color=color, label=f"r = {r_nm} nm")

    sl.plt.xlabel("Frequency (MHz)")
    sl.plt.ylabel("Normalized Intensity")
    sl.plt.title("Pake Patterns for Different Inter-Spin Distances")
    sl.plt.legend()
    sl.plt.grid(ls=":")
    sl.plt.show()



.. image-sg:: /sl_examples/images/sphx_glr_plot_09_PakePattern_002.png
   :alt: Pake Patterns for Different Inter-Spin Distances
   :srcset: /sl_examples/images/sphx_glr_plot_09_PakePattern_002.png
   :class: sphx-glr-single-img






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

   **Total running time of the script:** (0 minutes 0.656 seconds)


.. _sphx_glr_download_sl_examples_plot_09_PakePattern.py:

.. only:: html

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

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

      :download:`Download Jupyter notebook: plot_09_PakePattern.ipynb <plot_09_PakePattern.ipynb>`

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

      :download:`Download Python source code: plot_09_PakePattern.py <plot_09_PakePattern.py>`

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

      :download:`Download zipped: plot_09_PakePattern.zip <plot_09_PakePattern.zip>`


.. only:: html

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

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