LFC comparison

This example compares different static (and quasi-static) local field corrections that are available in jaxrts.

The model by Hubbard [Hubbard and Peierls, 1957] is a simple formula, and the first attempt to formulate a LFC, but does not obey the correct limiting cases.

The dotted curves showing the models by [Utsumi and Ichimaru, 1982] and [Farid et al., 1993] show no temperature dependence as they are only implemented for zero-temperature, whey they are the respective limits for the interpolation formulas.

Dornheim et al. [Dornheim et al., 2020] argue, that while the \(k^2\) scaling for large \(k\) as presented by Farid is correct for the static LFC, in practice, rather an effective static LFC should be used that obeys the correct limiting case for a full, \(\omega\) dependent LCF. An analytical function fitting their datapoints is provided in [Dornheim et al., 2021].

theta = 0.0069
theta = 3.5
theta = 6.9e+01

import jax.numpy as jnp
import jpu.numpy as jnpu
import matplotlib.pyplot as plt
import scienceplots  # noqa: F401

import jaxrts

ureg = jaxrts.ureg

plt.style.use("science")


temp = (
    jnp.array([0.1, 50, 1000]) * ureg.electron_volt / ureg.boltzmann_constant
)

n_e = 2.5e23 / ureg.centimeter**3
kf = jaxrts.plasma_physics.fermi_wavenumber(n_e)
k = jnpu.linspace(0 * kf, 7 * kf, 5000)

fig, ax = plt.subplots(len(temp), sharex=True, figsize=(6, 9))


for T, axis in zip(temp, ax):
    theta = (T / jaxrts.plasma_physics.fermi_temperature(n_e)).m_as(
        ureg.dimensionless
    )
    print(f"theta = {theta:.2}")
    if theta < 4:
        G_dornheim = [
            jaxrts.ee_localfieldcorrections.eelfc_dornheim2021(
                _k, T, n_e
            ).m_as(ureg.dimensionless)
            for _k in k
        ]
        axis.plot(
            (k / kf).m_as(ureg.dimensionless),
            G_dornheim,
            label="Dornheim.2021 (QSA)",
            color="C3",
        )
    else:
        axis.text(0, 2, "outside Dornheim.2021 validity", color="C3")
    G_hubbard = jaxrts.ee_localfieldcorrections.eelfc_hubbard(k, T, n_e)
    G_geldart = jaxrts.ee_localfieldcorrections.eelfc_geldartvosko(k, T, n_e)
    G_gregori = jaxrts.ee_localfieldcorrections.eelfc_interp_gregori2007(
        k, T, n_e
    )
    G_farid = jaxrts.ee_localfieldcorrections.eelfc_farid(k, T, n_e)
    G_utsumi = jaxrts.ee_localfieldcorrections.eelfc_utsumiichimaru(k, T, n_e)
    G_fortmann = jaxrts.ee_localfieldcorrections.eelfc_interp_fortmann2010(
        k, T, n_e
    )
    axis.plot(
        (k / kf).m_as(ureg.dimensionless),
        G_gregori.m_as(ureg.dimensionless),
        label="Interp. Gregori.2007",
    )
    axis.plot(
        (k / kf).m_as(ureg.dimensionless),
        G_fortmann.m_as(ureg.dimensionless),
        label="Interp. Fortmann.2010",
    )
    axis.plot(
        (k / kf).m_as(ureg.dimensionless),
        G_hubbard.m_as(ureg.dimensionless),
        label="Hubbard.1957",
    )
    axis.plot(
        (k / kf).m_as(ureg.dimensionless),
        G_utsumi.m_as(ureg.dimensionless),
        label="Utsumi \\& Ichimaru (zero T)",
        ls="dotted",
        color="gray",
    )
    axis.plot(
        (k / kf).m_as(ureg.dimensionless),
        G_farid.m_as(ureg.dimensionless),
        label="Farid (zero T)",
        ls="dotted",
        color="black",
    )
    axis.plot(
        (k / kf).m_as(ureg.dimensionless),
        G_geldart.m_as(ureg.dimensionless),
        label="Geldart \\& Vosko (high T)",
        ls="dashed",
        color="black",
    )
    axis.set_title(
        f"$k_BT=${(T * ureg.boltzmann_constant).m_as(ureg.electron_volt):.0e} eV"  # noqa:E501
    )
    axis.set_ylabel("$G$")

ax[0].legend(ncols=2)
ax[-1].set_xlabel("$k / k_f$")

plt.tight_layout()
plt.show()