def test_plasma_dispersion_deriv_errors(w, expected_error):
"""Test errors that should be raised by plasma_dispersion_func_deriv."""
with pytest.raises(expected_error):
plasma_dispersion_func_deriv(w)
pytest.fail(f"plasma_dispersion_func_deriv({w}) did not raise "
f"{expected_error.__name__} as expected.")
def test_plasma_dispersion_func_deriv(w, expected):
r"""Test plasma_dispersion_func_deriv against tabulated results
and an exact relationship."""
# The tabulated results are taken from Fried & Conte (1961). The
# exact analytical relationship comes from the bottom of page 3 of
# Fried & Conte (1961).
Z_deriv = plasma_dispersion_func_deriv(w)
assert np.isclose(Z_deriv, expected, atol=5e-5 * (1 + 1j), rtol=5e-6), (
f"The derivative of the plasma dispersion function does not match "
f"the expected value for w = {w}. The value of "
f"plasma_dispersion_func_deriv({w}) equals {Z_deriv} whereas the "
f"expected value is {expected}. The difference between the actual "
f"and expected results is {Z_deriv - expected}."
)
Z = plasma_dispersion_func(w)
Z_deriv_characterization = -2 * (1 + w * Z)
assert np.isclose(Z_deriv, Z_deriv_characterization, rtol=1e-15), (
f"The relationship that Z'(w) = -2 * [1 + w * Z(w)] is not "
f"met for w = {w}, where Z'(w) = {Z_deriv} and "
f"-2 * [1 + w * Z(w)] = {Z_deriv_characterization}."
)
for root in roots:
Z_at_root = plasma_dispersion_func(root)
assert np.isclose(Z_at_root, 0 + 0j, atol=1e-15 * (1 + 1j)), (
"A root of the plasma dispersion function is expected at w = "
f"{root}, but plasma_dispersion_func({root}) is equal to "
f"{Z_at_root} instead of {0j}."
)
# w, expected
plasma_disp_deriv_table = [
(0, -2),
(1, 0.152_318 - 1.304_10j),
(1j, -0.484_257),
(1.2 + 4.4j, -0.397_561e-1 - 0.217_392e-1j),
(9j, plasma_dispersion_func_deriv(9j * u.dimensionless_unscaled)),
(5.4 - 3.1j, 0.012_449_1 + 0.023_138_3j),
(9.9 - 10j, 476.153 + 553.121j),
(5 + 7j, -4.591_20e-3 - 0.012_610_4j),
(4.5 - 10j, 0.260_153e37 - 0.211_814e37j),
]
@pytest.mark.parametrize("w, expected", plasma_disp_deriv_table)
def test_plasma_dispersion_func_deriv(w, expected):
r"""Test plasma_dispersion_func_deriv against tabulated results
and an exact relationship."""
# The tabulated results are taken from Fried & Conte (1961). The
# exact analytical relationship comes from the bottom of page 3 of
# Fried & Conte (1961).
def permittivity_1D_Maxwellian(
omega: u.rad / u.s,
kWave: u.rad / u.m,
T: u.K,
n: u.m**-3,
particle,
z_mean: u.dimensionless_unscaled = None) -> u.dimensionless_unscaled:
r"""
The classical dielectric permittivity for a 1D Maxwellian plasma. This
function can calculate both the ion and electron permittivities. No
additional effects are considered (e.g. magnetic fields, relativistic
effects, strongly coupled regime, etc.)
Parameters
----------
omega : ~astropy.units.Quantity
The frequency in rad/s of the electromagnetic wave propagating
through the plasma.
kWave : ~astropy.units.Quantity
The corresponding wavenumber, in rad/m, of the electromagnetic wave
propagating through the plasma. This is often modulated by the
dispersion of the plasma or by relativistic effects. See em_wave.py
for ways to calculate this.
T : ~astropy.units.Quantity
The plasma temperature - this can be either the electron or the ion
temperature, but should be consistent with density and particle.
n : ~astropy.units.Quantity
The plasma density - this can be either the electron or the ion
density, but should be consistent with temperature and particle.
particle : str
The plasma particle species.
z_mean : str
The average ionization of the plasma. This is only required for
calculating the ion permittivity.
Returns
-------
chi : ~astropy.units.Quantity
The ion or the electron dielectric permittivity of the plasma.
This is a dimensionless quantity.
Notes
-----
The dielectric permittivities for a Maxwellian plasma are described
by the following equations [1]_
.. math::
\chi_e(k, \omega) = - \frac{\alpha_e^2}{2} Z'(x_e)
\chi_i(k, \omega) = - \frac{\alpha_i^2}{2}\frac{Z}{} Z'(x_i)
\alpha = \frac{\omega_p}{k v_{Th}}
x = \frac{\omega}{k v_{Th}}
:math:`chi_e` and :math:`chi_i` are the electron and ion permittivities
respectively. :math:`Z'` is the derivative of the plasma dispersion
function. :math:`\alpha` is the scattering parameter which delineates
the difference between the collective and non-collective Thomson
scattering regimes. :math:`x` is the dimensionless phase velocity
of the EM wave propagating through the plasma.
References
----------
.. [1] J. Sheffield, D. Froula, S. H. Glenzer, and N. C. Luhmann Jr,
Plasma scattering of electromagnetic radiation: theory and measurement
techniques. Chapter 5 Pg 106 (Academic press, 2010).
Example
-------
>>> from astropy import units as u
>>> from numpy import pi
>>> from astropy.constants import c
>>> T = 30 * 11600 * u.K
>>> n = 1e18 * u.cm**-3
>>> particle = 'Ne'
>>> z_mean = 8 * u.dimensionless_unscaled
>>> vTh = parameters.thermal_speed(T, particle, method="most_probable")
>>> omega = 5.635e14 * 2 * pi * u.rad / u.s
>>> kWave = omega / vTh
>>> permittivity_1D_Maxwellian(omega, kWave, T, n, particle, z_mean)
<Quantity -6.72809...e-08+5.76037...e-07j>
"""
# thermal velocity
vTh = parameters.thermal_speed(T=T,
particle=particle,
method="most_probable")
# plasma frequency
wp = parameters.plasma_frequency(n=n, particle=particle, z_mean=z_mean)
# scattering parameter alpha.
# explicitly removing factor of sqrt(2) to be consistent with Froula
alpha = np.sqrt(2) * (wp / (kWave * vTh)).to(u.dimensionless_unscaled)
# The dimensionless phase velocity of the propagating EM wave.
zeta = (omega / (kWave * vTh)).to(u.dimensionless_unscaled)
chi = alpha**2 * (-1 / 2) * plasma_dispersion_func_deriv(zeta.value)
return chi
