.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "api/gallery/spectro/rotational/plot_01_rotational_spectrum.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_rotational_plot_01_rotational_spectrum.py>`
        to download the full example code.

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

.. _sphx_glr_api_gallery_spectro_rotational_plot_01_rotational_spectrum.py:


Dennison's quantized rigid rotor: HCl rotational lines spaced by 2B, with Boltzmann intensities
================================================================================================

Dennison (1926) gave the quantum-mechanical rotational energy levels
:math:`E_J=\hbar^2J(J+1)/(2I)`. With the :math:`\Delta J=+1` selection
rule, these levels produce absorption lines at :math:`2B(J+1)`, which
are evenly spaced by :math:`2B`. This example builds an HCl-like rigid
rotor from :mod:`chemistrykit.quantum`, prints its level energies and
line positions, checks the constant :math:`2B` spacing, and adds the
Boltzmann-population intensity pattern seen in a real far-infrared or
microwave spectrum.

.. GENERATED FROM PYTHON SOURCE LINES 16-31

.. code-block:: Python

    import matplotlib.pyplot as plt
    import numpy as np
    import scipy.constants as sc

    from chemistrykit.quantum.systems.rigid_rotor import RigidRotor
    from chemistrykit.spectro.systems.rotational import energy_to_wavenumber, rotational_line_wavenumbers, rotational_spectrum
    from chemistrykit.spectro.visualizers.spectro_plots import plot_broadened_spectrum, plot_stick_spectrum

    rotor_hcl = RigidRotor.from_diatomic(mass1=1.008 * sc.atomic_mass, mass2=34.97 * sc.atomic_mass, bond_length=127.5e-12)

    levels = np.array([energy_to_wavenumber(rotor_hcl.energy(J)) for J in range(6)])
    print(f"Rotational levels E_J/hc (cm^-1), J=0..5: {np.round(levels, 2)}")
    B = levels[1] / 2.0  # E_1 = 2B
    print(f"Rotational constant B = {B:.3f} cm^-1; E_J / B = {np.round(levels / B, 3)} = J(J+1)")





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

 .. code-block:: none

    Rotational levels E_J/hc (cm^-1), J=0..5: [  0.    21.17  63.51 127.01 211.68 317.53]
    Rotational constant B = 10.584 cm^-1; E_J / B = [ 0.  2.  6. 12. 20. 30.] = J(J+1)




.. GENERATED FROM PYTHON SOURCE LINES 32-35

Transitions :math:`J\to J+1` fall at :math:`2B(J+1)`, which gives an
evenly spaced line pattern, even though the levels themselves are not
evenly spaced:

.. GENERATED FROM PYTHON SOURCE LINES 35-40

.. code-block:: Python


    lines_hcl = rotational_line_wavenumbers(rotor_hcl, J_max=8)
    print(f"HCl lines (cm^-1): {np.round(lines_hcl, 2)}")
    print(f"Line spacings (cm^-1): {np.round(np.diff(lines_hcl), 3)}  (2B = {2 * B:.3f})")





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

 .. code-block:: none

    HCl lines (cm^-1): [ 21.17  42.34  63.51  84.67 105.84 127.01 148.18 169.35 190.52]
    Line spacings (cm^-1): [21.168 21.168 21.168 21.168 21.168 21.168 21.168 21.168]  (2B = 21.168)




.. GENERATED FROM PYTHON SOURCE LINES 41-44

The full spectrum, with relative intensities from the Boltzmann
population and :math:`(2J+1)` degeneracy of each initial J level. The
strongest line comes from the most populated level, not from J=0:

.. GENERATED FROM PYTHON SOURCE LINES 44-58

.. code-block:: Python


    spectrum = rotational_spectrum(rotor_hcl, J_max=15, temperature=300.0)
    fig, (ax1, ax2) = plt.subplots(1, 2, figsize=(11, 4))
    plot_stick_spectrum(spectrum, ax=ax1, color="steelblue")
    ax1.set_xlabel(r"wavenumber (cm$^{-1}$)")
    ax1.set_title("HCl rotational spectrum (stick, 300 K)")

    x = np.linspace(0, spectrum.positions[-1] + 20.0, 2000)
    plot_broadened_spectrum(spectrum, x, ax=ax2, shape="lorentzian", fwhm=3.0, color="crimson")
    ax2.set_xlabel(r"wavenumber (cm$^{-1}$)")
    ax2.set_title("Lorentzian-broadened spectrum")

    fig.tight_layout()
    plt.show()



.. image-sg:: /api/gallery/spectro/rotational/images/sphx_glr_plot_01_rotational_spectrum_001.png
   :alt: HCl rotational spectrum (stick, 300 K), Lorentzian-broadened spectrum
   :srcset: /api/gallery/spectro/rotational/images/sphx_glr_plot_01_rotational_spectrum_001.png
   :class: sphx-glr-single-img






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

   **Total running time of the script:** (0 minutes 0.065 seconds)


.. _sphx_glr_download_api_gallery_spectro_rotational_plot_01_rotational_spectrum.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_rotational_spectrum.ipynb <plot_01_rotational_spectrum.ipynb>`

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

      :download:`Download Python source code: plot_01_rotational_spectrum.py <plot_01_rotational_spectrum.py>`

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

      :download:`Download zipped: plot_01_rotational_spectrum.zip <plot_01_rotational_spectrum.zip>`


.. only:: html

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

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