A global model of an argon discharge: electron temperature and density against pressure.

The textbook global model (Lieberman and Lichtenberg, ch. 10): one ion species, ionization from a step cross section averaged over a Maxwellian (with plasmacoll), elastic losses, ions leaving at the Bohm speed. The particle balance alone sets the electron temperature, which falls with the pressure; the power balance then sets the density. Magnetic cusps that cover most of the wall lower both the losses and the electron temperature. The cross sections are illustrative step functions, not argon data. Every scan is one xarray dataset (:func:wellbalanced.stack), plotted with plasma-plots.

Python source

"""A global model of an argon discharge: electron temperature and density against pressure.

The textbook global model (Lieberman and Lichtenberg, ch. 10): one ion species,
ionization from a step cross section averaged over a Maxwellian (with plasmacoll),
elastic losses, ions leaving at the Bohm speed. The particle balance alone sets the
electron temperature, which falls with the pressure; the power balance then sets the
density. Magnetic cusps that cover most of the wall lower both the losses and the
electron temperature. The cross sections are illustrative step functions, not
argon data. Every scan is one xarray dataset (:func:`wellbalanced.stack`), plotted
with plasma-plots.
"""

import numpy as np
import plasma_plots
import xarray as xr
from plasmacoll import CrossSection
from plasmacoll.constants import BOLTZMANN, ELECTRON_MASS

from wellbalanced import ELECTRON, Chamber, Cusp, GlobalModel, Reaction, Species, stack

argon = Species.atomic("Ar", 39.948)
argon_ion = Species.atomic("Ar+", 39.948, 1)
reactions = [
    Reaction.from_cross_section(
        ("e", "Ar"),
        ("e", "e", "Ar+"),
        CrossSection.constant(2.0e-20, threshold=15.76),
        ELECTRON_MASS,
        electron_energy_loss=15.76,
        name="ionization",
    ),
    Reaction.from_cross_section(
        ("e", "Ar"),
        ("e", "Ar"),
        CrossSection.constant(1.0e-20, threshold=11.5),
        ELECTRON_MASS,
        electron_energy_loss=11.5,
        name="excitation",
    ),
    Reaction(
        ("e", "Ar"), ("e", "Ar"), 1.0e-13, elastic_mass=argon.mass, name="elastic"
    ),
]
# a cylinder of 10 cm radius and 20 cm length
radius, length = 0.1, 0.2
volume = np.pi * radius**2 * length
wall = 2 * np.pi * radius * (radius + length)
chambers = {
    "open walls": Chamber(volume, wall, h_factor=0.3),
    "cusps on the side": Chamber(
        volume,
        wall,
        ion_loss_area=2 * np.pi * radius**2,
        h_factor=0.3,
        cusp=Cusp(field=0.1, line_length=16 * length),
    ),
}

pressures = np.geomspace(0.1, 10.0, 15)  # Pa
gas = pressures / (BOLTZMANN * 300.0)  # m^-3
scans = []
for chamber in chambers.values():
    model = GlobalModel(
        species=[ELECTRON, argon, argon_ion],
        reactions=reactions,
        chamber=chamber,
        power=500.0,
        fixed={"Ar": float(gas[0])},
    )
    states = model.scan("fixed", [{"Ar": float(n)} for n in gas])
    scans.append(stack(states, "pressure", pressures, units="Pa"))
result = xr.concat(scans, dim="case").assign_coords(case=list(chambers))

with plasma_plots.figure(1, 2, title="Argon, 500 W absorbed") as fig:
    result.electron_temperature.plasma.plot.profiles(
        x="pressure", over="case", logx=True, ax=fig[0]
    )
    result.density.sel(species="e").plasma.plot.profiles(
        x="pressure",
        over="case",
        logx=True,
        logy=True,
        title="electron density",
        ax=fig[1],
    )

Plots