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

.. only:: html

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

        :ref:`Go to the end <sphx_glr_download_gen_examples_analysis_plot_f-sum-rule.py>`
        to download the full example code

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

.. _sphx_glr_gen_examples_analysis_plot_f-sum-rule.py:


f-sum rule for the dynamic stucture factor
==========================================

This example shows the computation of the f-sum rule for an example spectrum as
suggested by :cite:`Dornheim.2024`.
If fully ionized, we see excellent agreement for the RPA, and also the
Born-Mermin approximation for the free-free scattering yields in reasonable
results.
The implemented bound-free Model, however, does violate the f-sum rule. One
could leverage this to fix a value for
:py:attr:`jaxrts.models.SchumacherImpulse.r_k`, so that the f-sum rule would be
fulfilled.

.. GENERATED FROM PYTHON SOURCE LINES 15-94



.. image-sg:: /gen_examples/analysis/images/sphx_glr_plot_f-sum-rule_001.svg
   :alt: plot f sum rule
   :srcset: /gen_examples/analysis/images/sphx_glr_plot_f-sum-rule_001.svg
   :class: sphx-glr-single-img





.. code-block:: Python


    from functools import partial

    import jax
    import matplotlib.pyplot as plt
    from jax import numpy as jnp

    import jaxrts

    ureg = jaxrts.ureg

    plt.style.use("science")

    state = jaxrts.PlasmaState(
        ions=[jaxrts.Element("N")],
        Z_free=jnp.array([7]),
        mass_density=jnp.array([1]) * ureg.gram / ureg.centimeter**3,
        T_e=40 * ureg.electron_volt / ureg.k_B,
    )
    setup = jaxrts.Setup(
        scattering_angle=ureg("60°"),
        energy=ureg("9 keV"),
        measured_energy=ureg("9 keV")
        + jnp.linspace(-2000, 2000, 5000) * ureg.electron_volt,
        instrument=partial(
            jaxrts.instrument_function.instrument_gaussian,
            sigma=ureg("5eV") / ureg.hbar / (2 * jnp.sqrt(2 * jnp.log(2))),
        ),
    )

    # Turn off the dispersion correction
    setup.correct_k_dispersion = False

    state["screening length"] = jaxrts.models.ArbitraryDegeneracyScreeningLength()
    state["screening"] = jaxrts.models.LinearResponseScreening()
    state["ionic scattering"] = jaxrts.models.OnePotentialHNCIonFeat()
    # state["free-free scattering"] = jaxrts.models.RPA_DandreaFit()
    state["free-free scattering"] = jaxrts.models.BornMermin_Fit()
    state["BM S_ii"] = jaxrts.models.Sum_Sii()
    state["bound-free scattering"] = jaxrts.models.Neglect()
    state["free-bound scattering"] = jaxrts.models.DetailedBalance()


    S_ee = state.probe(setup)

    # x is the cut-off Energy
    x = jnp.linspace(1, 1800) * ureg.electron_volt

    fsum = jax.vmap(
        jaxrts.analysis.ITCF_fsum,
        in_axes=(None, None, 0),
        out_axes=(0),
    )(S_ee, setup, x)

    fig, ax = plt.subplots()
    ax.plot(
        x.m_as(ureg.electron_volt),
        (fsum).m_as(ureg.electron_volt),
        label="auto",
    )

    ax.hlines(
        [
            ((-1) * ureg.hbar**2 * setup.k**2 / (2 * ureg.electron_mass)).m_as(
                ureg.electron_volt
            )
        ],
        *ax.get_xlim(),
        color="gray",
        ls="dashed",
    )


    ax.set_ylabel(r"$\frac{\partial F}{\partial \tau}\vert_{\tau=0}$ [eV]")
    ax.set_xlabel("$x$ [eV]")
    ax.legend()
    fig.tight_layout()

    plt.show()


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

   **Total running time of the script:** (0 minutes 31.005 seconds)


.. _sphx_glr_download_gen_examples_analysis_plot_f-sum-rule.py:

.. only:: html

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

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

      :download:`Download Jupyter notebook: plot_f-sum-rule.ipynb <plot_f-sum-rule.ipynb>`

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

      :download:`Download Python source code: plot_f-sum-rule.py <plot_f-sum-rule.py>`


.. only:: html

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

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