.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "api/gallery/spectro/lineshapes/plot_01_lineshapes.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_spectro_lineshapes_plot_01_lineshapes.py>`
        to download the full example code.

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

.. _sphx_glr_api_gallery_spectro_lineshapes_plot_01_lineshapes.py:


Gaussian, Lorentzian, and Voigt lineshapes: two broadening mechanisms and their convolution
================================================================================================

Compares the three standard spectral lineshapes at matched FWHM: a
Lorentzian (the shape a classically radiating damped oscillator, or a
finite-lifetime quantum transition, produces -- "homogeneous"
broadening, identical for every molecule in the sample), a Gaussian (the
shape a Maxwell-Boltzmann spread of line-of-sight velocities produces via
the Doppler effect -- "inhomogeneous" broadening, different molecules
contributing different, momentarily Doppler-shifted, sub-lines), and
their convolution, the Voigt profile, which is what a real spectrum
typically shows once both mechanisms contribute together. The Lorentzian
has visibly heavier tails than the Gaussian at the same FWHM -- the
qualitative signature used to diagnose which mechanism dominates a real
measured line.

.. GENERATED FROM PYTHON SOURCE LINES 20-42

.. code-block:: Python

    import matplotlib.pyplot as plt
    import numpy as np

    from chemistrykit.spectro.utils.lineshapes import gaussian, lorentzian, voigt

    fwhm = 4.0  # shared FWHM, arbitrary wavenumber-like units
    x = np.linspace(-20.0, 20.0, 2001)

    g = gaussian(x, x0=0.0, fwhm=fwhm)
    lorentz = lorentzian(x, x0=0.0, fwhm=fwhm)
    v = voigt(x, x0=0.0, fwhm_gaussian=fwhm, fwhm_lorentzian=fwhm)

    print(f"Peak height -- Gaussian: {np.max(g):.4f}, Lorentzian: {np.max(lorentz):.4f}, Voigt: {np.max(v):.4f}")

    # All three are normalized to unit area (over a wide enough grid to
    # capture the Lorentzian's slowly-decaying tails):
    x_wide = np.linspace(-2000.0, 2000.0, 400_001)
    area_g = np.trapezoid(gaussian(x_wide, 0.0, fwhm), x_wide)
    area_l = np.trapezoid(lorentzian(x_wide, 0.0, fwhm), x_wide)
    area_v = np.trapezoid(voigt(x_wide, 0.0, fwhm, fwhm), x_wide)
    print(f"Areas (Gaussian, Lorentzian, Voigt): {area_g:.4f}, {area_l:.4f}, {area_v:.4f}")





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

 .. code-block:: none

    Peak height -- Gaussian: 0.2349, Lorentzian: 0.1592, Voigt: 0.1123
    Areas (Gaussian, Lorentzian, Voigt): 1.0000, 0.9994, 0.9994




.. GENERATED FROM PYTHON SOURCE LINES 43-49

The Lorentzian's heavier tails, at exactly matched FWHM, are the
textbook way to tell homogeneous (lifetime/collisional) broadening
apart from inhomogeneous (Doppler) broadening in a measured spectrum --
far from line center the Lorentzian sits orders of magnitude above the
Gaussian, even though both have fallen to the same half-maximum at
+/-fwhm/2:

.. GENERATED FROM PYTHON SOURCE LINES 49-54

.. code-block:: Python


    far_from_center = np.abs(x) > 3.0 * fwhm
    ratio = np.mean(lorentz[far_from_center] / g[far_from_center])
    print(f"\nAt |x| > 3*FWHM, the Lorentzian exceeds the Gaussian by a factor of order 1e{np.log10(ratio):.0f}")





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

 .. code-block:: none


    At |x| > 3*FWHM, the Lorentzian exceeds the Gaussian by a factor of order 1e26




.. GENERATED FROM PYTHON SOURCE LINES 55-67

.. code-block:: Python

    fig, ax = plt.subplots(figsize=(7, 5))
    ax.plot(x, g, label="Gaussian (Doppler / inhomogeneous)", color="steelblue")
    ax.plot(x, lorentz, label="Lorentzian (lifetime / homogeneous)", color="crimson")
    ax.plot(x, v, "--", label="Voigt (convolution of both)", color="black")
    ax.set_yscale("log")
    ax.set_ylim(1e-5, 1.0)
    ax.set_xlabel("x - x0 (arbitrary units)")
    ax.set_ylabel("intensity (log scale)")
    ax.set_title(f"Lineshapes at matched FWHM = {fwhm:.0f}")
    ax.legend()
    fig.tight_layout()
    plt.show()



.. image-sg:: /api/gallery/spectro/lineshapes/images/sphx_glr_plot_01_lineshapes_001.png
   :alt: Lineshapes at matched FWHM = 4
   :srcset: /api/gallery/spectro/lineshapes/images/sphx_glr_plot_01_lineshapes_001.png
   :class: sphx-glr-single-img






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

   **Total running time of the script:** (0 minutes 0.101 seconds)


.. _sphx_glr_download_api_gallery_spectro_lineshapes_plot_01_lineshapes.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_lineshapes.ipynb <plot_01_lineshapes.ipynb>`

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

      :download:`Download Python source code: plot_01_lineshapes.py <plot_01_lineshapes.py>`

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

      :download:`Download zipped: plot_01_lineshapes.zip <plot_01_lineshapes.zip>`


.. only:: html

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

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