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

.. only:: html

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

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

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

.. _sphx_glr_auto_gallery_accuracy_plot_density_scaling.py:


Linear density scaling against Lorentz-Lorenz
=============================================

The refractive index of a gas at number density :math:`N` follows from the
per-particle polarizability :math:`\alpha` through the Lorentz-Lorenz relation. The
linear scaling :math:`\chi = N\alpha/\varepsilon_0` is
its dilute limit. This figure shows the relative error of linear scaling in
:math:`\chi = n^2 - 1` against pressure, for a polarizability typical of each gas.

The polarizabilities are illustrative round numbers (atomic units), not curated data:
the error depends only on :math:`\chi` itself, :math:`1/(1-y) - 1` with
:math:`y = N\alpha/(3\varepsilon_0)`.

.. GENERATED FROM PYTHON SOURCE LINES 21-44



.. image-sg:: /auto_gallery/accuracy/images/sphx_glr_plot_density_scaling_001.svg
   :alt: plot density scaling
   :srcset: /auto_gallery/accuracy/images/sphx_glr_plot_density_scaling_001.svg
   :class: sphx-glr-single-img





.. code-block:: Python

    import matplotlib.pyplot as plt
    import numpy as np

    from optmat.conditions import State
    from optmat.constants import AU_POLARIZABILITY, BAR
    from optmat.gas.eos import number_density
    from optmat.gas.lorentz_lorenz import susceptibility, susceptibility_dilute

    pressures = np.geomspace(0.1, 150.0, 60)  # bar
    fig, ax = plt.subplots(figsize=(6, 4), layout="constrained")
    for gas, alpha_au in [("He", 1.4), ("Ar", 11.1), ("Xe", 27.3)]:
        errors = []
        for p in pressures:
            density = number_density(gas, State(p * BAR, 293.15))
            alpha = alpha_au * AU_POLARIZABILITY
            exact = susceptibility(alpha, density)
            errors.append(1 - susceptibility_dilute(alpha, density) / exact)
        ax.loglog(pressures, errors, label=f"{gas} (alpha = {alpha_au} a.u.)")
    ax.set_xlabel("pressure (bar), 20 °C, real gas")
    ax.set_ylabel("relative error of linear scaling in χ")
    ax.legend()
    ax.grid(which="both", alpha=0.3)
    plt.show()


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

   **Total running time of the script:** (0 minutes 0.437 seconds)


.. _sphx_glr_download_auto_gallery_accuracy_plot_density_scaling.py:

.. only:: html

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

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

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

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

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

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

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