"""
Functionality for calculating Coulomb parameters for different
configurations. Including a number of functions for handling Coulomb
collisions spanning weakly coupled (low density) to strongly coupled
(high density) regimes.
Coulomb collisions are collisions where the interaction force is
conveyed via the electric field, instead of any kind of contact
force. They usually result in relatively small deflections of particle
trajectories. However, given that there are many charged particles in a
plasma, one has to take into account the cumulative effects of many such
collisions.
"""
__all__ = [
"Coulomb_logarithm",
"Coulomb_cross_section",
]
import warnings
from numbers import Real
import astropy.units as u
import numpy as np
from plasmapy import particles
from plasmapy.formulary.collisions import lengths
from plasmapy.utils.decorators import validate_quantities
from plasmapy.utils.exceptions import CouplingWarning
[docs]
@validate_quantities(
T={"can_be_negative": False, "equivalencies": u.temperature_energy()},
V={"none_shall_pass": True},
)
@particles.particle_input
def Coulomb_logarithm(
T: u.Quantity[u.K],
n_e: u.Quantity[u.m**-3],
species: (particles.Particle, particles.Particle),
z_mean: Real = np.nan,
V: u.Quantity[u.m / u.s] = np.nan * u.m / u.s,
method="classical",
):
r"""
Compute the Coulomb logarithm.
Parameters
----------
T : `~astropy.units.Quantity`
Temperature in units of temperature or energy per particle,
which is assumed to be equal for both the test particle and the
target particle.
n_e : `~astropy.units.Quantity`
The electron number density in units convertible to m\ :sup:`-3`.
species : `tuple`
A tuple containing string representations of the test particle
(listed first) and the target particle (listed second).
z_mean : `~astropy.units.Quantity`, optional
The average ionization (arithmetic mean) of a plasma for which a
macroscopic description is valid. This parameter is used to
compute the average ion density (given the average ionization
and electron density) for calculating the ion sphere radius for
non-classical impact parameters. ``z_mean`` is a required
parameter if ``method`` is ``"ls_full_interp"``,
``"hls_max_interp"``, or ``"hls_full_interp"``.
V : `~astropy.units.Quantity`, optional
The relative velocity between particles. If not provided,
thermal velocity is assumed: :math:`μ V^2 \sim 2 k_B T` where
:math:`μ` is the reduced mass.
method : `str`, optional
The method by which to compute the Coulomb logarithm. The
default method is the classical straight-line Landau-Spitzer
method (``"classical"`` or ``"ls"``). The other six supported
methods are ``"ls_min_interp"``, ``"ls_full_interp"``,
``"ls_clamp_mininterp"``, ``"hls_min_interp"``,
``"hls_max_interp"``, and ``"hls_full_interp"``. Please refer
to the Notes section of this docstring for more information,
including about abbreviated aliases of these names.
Returns
-------
ln_Lambda : `float` or `numpy.ndarray`
The dimensionless Coulomb logarithm.
Raises
------
`ValueError`
If the mass or charge of either particle cannot be found, or any
of the inputs contain incorrect values.
`~astropy.units.UnitConversionError`
If the units on any of the inputs are incorrect, or if any of
``n_e``, ``T``, or ``V`` is not a `~astropy.units.Quantity`.
`~plasmapy.utils.exceptions.PhysicsError`
If the result is smaller than 1.
`~plasmapy.utils.exceptions.RelativityError`
If the input velocity is same or greater than the speed of
light.
Warns
-----
: `~astropy.units.UnitsWarning`
If units are not provided, SI units are assumed.
: `~plasmapy.utils.exceptions.RelativityWarning`
If the input velocity is greater than 5% of the speed of light.
Notes
-----
**Summary of Supported Methods of Computing the Coulomb Logarithm**
PlasmaPy supports seven methods of computing the Coulomb logarithm:
1. ``"classical"`` or ``"ls"``
2. ``"ls_min_interp"`` or ``"GMS-1"``
3. ``"ls_full_interp"`` or ``"GMS-2"``
4. ``"ls_clamp_mininterp"`` or ``"GMS-3"``
5. ``"hls_min_interp"`` or ``"GMS-4"``
6. ``"hls_max_interp"`` or ``"GMS-5"``
7. ``"hls_full_interp"`` or ``"GMS-6"``
Options 1–4 are straight-line Landau-Spitzer (``"ls..."``) methods
in which the trajectory of a Coulomb collision is modeled as a
straight line. For the straight-line Landau-Spitzer methods, the
Coulomb logarithm (:math:`\ln{Λ}`) is defined to be:
.. math::
\ln{Λ} \equiv \ln\left( \frac{b_{max}}{b_{min}} \right)
Options 5–7 are hyperbolic Landau-Spitzer (``"hls..."``) methods in
which the trajectory of a Coulomb collision is modeled as a
hyperbola. For the hyperbolic Landau-Spitzer methods, the Coulomb
logarithm (:math:`\ln{Λ}`) is defined to be:
.. math::
\ln{Λ} \equiv \frac{1}{2} \ln\left(1 + \frac{b_{max}^2}{b_{min}^2} \right)
For all 7 methods, :math:`b_{min}` and :math:`b_{max}` are the inner
impact parameter and the outer impact parameter, respectively, for
Coulomb collisions :cite:p:`spitzer:1962`\ ; :math:`b_{min}` and
:math:`b_{max}` are each computed by
`~plasmapy.formulary.collisions.lengths.impact_parameter`.
The abbreviations of Options 2–7 (``"GMS-..."``) refer to the first
initials of the three authors of :cite:t:`gericke:2002`.
.. note::
For strongly-coupled plasma, PlasmaPy recommends Option 7,
``"hls_full_interp"`` or ``"GMS-6"``, because of its high
accuracy regardless of a plasma's strength of coupling.
**Explanation of Supported Methods of Computing the Coulomb Logarithm**
In this section, further information about each method, such as
about interpolation and other special features, is documented.
Please refer to :cite:t:`spitzer:1962` and :cite:t:`gericke:2002`
for additional information about these methods.
Option 1: ``"classical"`` or ``"ls"`` (Landau-Spitzer)
The classical straight-line Landau-Spitzer method in which
:math:`b_{min}` is defined to be the higher of the de Broglie
wavelength (:math:`λ_{de Broglie}`) and the distance of closest
approach (:math:`ρ_⟂`) if they are not equal (and either of the
two if they are equal) and :math:`b_{max}` is defined to be the
Debye length (:math:`λ_{Debye}`).
.. math::
\ln{Λ} \equiv \ln\left( \frac{b_{max}}{b_{min}} \right)
.. math::
b_{min} \equiv
\left\{
\begin{array}{ll}
λ_{de Broglie} & \mbox{if } λ_{de Broglie} \geq ρ_⟂ \\
ρ_⟂ & \mbox{if } ρ_⟂ \geq λ_{de Broglie}
\end{array}
\right.
.. math::
b_{max} \equiv λ_{Debye}
The inner impact parameter (:math:`b_{min}`) is the higher of
:math:`λ_{de Broglie}` and :math:`ρ_⟂` because for impact
parameters lower than :math:`λ_{de Broglie}`, quantum effects
cause the collision to be non-Coulombic
:cite:p:`chen:2016,fundamenski:2007`.
The outer impact parameter (:math:`b_{max}`) is defined to be
the Debye length (:math:`λ_{Debye}`) because at distances higher
than the Debye length, the electric fields created by other
particles are screened out by the electrons rearranging
themselves.
The uncertainty of the classical straight-line Landau-Spitzer
method is on the order of its reciprocal.
This method is invalid if :math:`\ln{Λ} < 2` because of the
uncertainty of this method and is invalid if :math:`\ln{Λ} < 0`,
which may be true if the coupling parameter is high (such as for
nonideal, dense, cold plasmas).
Please refer to :cite:t:`spitzer:1962` for additional
information about this method.
Option 2: ``"ls_min_interp"`` or ``"GMS-1"`` (Landau-Spitzer,
interpolation of :math:`b_{min}`)
A straight-line Landau-Spitzer method in which :math:`b_{min}`
is interpolated between the de Broglie wavelength (:math:`λ_{de
Broglie}`) and the distance of closest approach (:math:`ρ_⟂`)
and :math:`b_{max}` is defined to be the Debye length
(:math:`λ_{Debye}`).
.. math::
\ln{Λ} \equiv \ln\left( \frac{b_{max}}{b_{min}} \right)
.. math::
b_{min} \equiv \sqrt{λ_{de Broglie}^2 + ρ_⟂^2}
.. math::
b_{max} \equiv λ_{Debye}
This method is invalid if :math:`\ln{Λ} < 0`, which may be true
if the coupling parameter is high (such as for nonideal, dense,
cold plasmas).
Note: This is the first method in Table 1 of
:cite:t:`gericke:2002`.
Option 3: ``"ls_full_interp"`` or ``"GMS-2"`` (Landau-Spitzer,
interpolation of :math:`b_{min}` and :math:`b_{max}`)
A straight-line Landau-Spitzer method in which :math:`b_{min}`
and :math:`b_{max}` are each interpolated. :math:`b_{min}` is
interpolated between the de Broglie wavelength (:math:`λ_{de
Broglie}`) and the distance of closest approach (:math:`ρ_⟂`).
:math:`b_{max}` is interpolated between the Debye length
(:math:`λ_{Debye}`) and the ion sphere radius (:math:`a_i`).
.. math::
\ln{Λ} \equiv \ln\left( \frac{b_{max}}{b_{min}} \right)
.. math::
b_{min} \equiv \sqrt{λ_{de Broglie}^2 + ρ_⟂^2}
.. math::
b_{max} \equiv \sqrt{λ_{Debye}^2 + a_i^2}
This method is invalid if :math:`\ln{Λ} < 0`, which may be true
if the coupling parameter is high (such as for nonideal, dense,
cold plasmas).
Note: This is the second method in Table 1 of
:cite:t:`gericke:2002`.
Option 4: ``"ls_clamp_mininterp"`` or ``"GMS-3"`` (Landau-Spitzer
with a clamp, interpolation of :math:`b_{min}`)
A straight-line Landau-Spitzer method in which the value of
:math:`\ln{Λ}` is clamped at a minimum of :math:`2` so that it
is impossible for :math:`\ln{Λ} < 0` (which is possible by the
classical Landau-Spitzer method). :math:`b_{min}` is
interpolated between the de Broglie wavelength (:math:`λ_{de
Broglie}`) and the distance of closest approach
(:math:`ρ_⟂`). :math:`b_{max}` is defined to be the Debye length
(:math:`λ_{Debye}`).
.. math::
\ln{Λ} \equiv
\left\{
\begin{array}{ll}
\ln\left(\frac{b_{max}}{b_{min}} \right)
& \mbox{if } \ln\left( \frac{b_{max}}{b_{min}}\right)
\geq 2 \\
2 & \mbox{if } \ln\left( \frac{b_{max}}{b_{min}} \right)
\leq 2
\end{array}
\right.
.. math::
b_{min} \equiv \sqrt{λ_{de Broglie}^2 + ρ_⟂^2}
.. math::
b_{max} \equiv λ_{Debye}
This method is valid for any plasma because it is impossible for
:math:`\ln{Λ} < 0` by this method, even if the coupling
parameter is high (such as for nonideal, dense, cold plasmas).
Note: This is the third method in Table 1 of
:cite:t:`gericke:2002`.
Option 5: ``"hls_min_interp"`` or ``"GMS-4"`` (Hyperbolic
Landau-Spitzer, interpolation of :math:`b_{min}`)
A hyperbolic Landau-Spitzer method in which :math:`b_{min}` is
interpolated between the de Broglie wavelength (:math:`λ_{de
Broglie}`) and the distance of closest approach (:math:`ρ_⟂`)
and :math:`b_{max}` is defined to be the Debye length
(:math:`λ_{Debye}`).
.. math::
\ln{Λ} \equiv \frac{1}{2} \ln\left(1 +
\frac{b_{max}^2}{b_{min}^2} \right)
.. math::
b_{min} \equiv \sqrt{λ_{de Broglie}^2 + ρ_⟂^2}
.. math::
b_{max} \equiv λ_{Debye}
This method is valid for any plasma because it is impossible for
:math:`\ln{Λ} < 0` by this method, even if the coupling
parameter is high (such as for nonideal, dense, cold plasmas).
Note: This is the fourth method in Table 1 of
:cite:t:`gericke:2002`.
Option 6: ``"hls_max_interp"`` or ``"GMS-5"`` (Hyperbolic
Landau-Spitzer, interpolation of :math:`b_{max}`)
A hyperbolic Landau-Spitzer method in which :math:`b_{max}` is
interpolated between the Debye length (:math:`λ_{Debye}`) and
the ion sphere radius (:math:`a_i`) and :math:`b_{min}` is
defined to be the distance of closest approach (:math:`ρ_⟂`).
.. math::
\ln{Λ} \equiv \frac{1}{2} \ln\left(1 +
\frac{b_{max}^2}{b_{min}^2} \right)
.. math::
b_{min} \equiv ρ_⟂
.. math::
b_{max} \equiv \sqrt{λ_{Debye}^2 + a_i^2}
This method is valid for any plasma because it is impossible for
:math:`\ln{Λ} < 0` by this method, even if the coupling
parameter is high (such as for nonideal, dense, cold plasmas).
Caution: This method overestimates :math:`\ln{Λ}` at high
temperatures.
Note: This is the fifth method in Table 1 of
:cite:t:`gericke:2002`.
Option 7: ``"hls_full_interp"`` or ``"GMS-6"`` (Hyperbolic
Landau-Spitzer, interpolation of :math:`b_{min}` and
:math:`b_{max}`)
A hyperbolic Landau-Spitzer method in which :math:`b_{min}` and
:math:`b_{max}` are each interpolated. :math:`b_{min}` is
interpolated between the de Broglie wavelength (:math:`λ_{de
Broglie}`) and the distance of closest approach (:math:`ρ_⟂`).
:math:`b_{max}` is interpolated between the Debye length
(:math:`λ_{Debye}`) and the ion sphere radius (:math:`a_i`).
.. math::
\ln{Λ} \equiv \frac{1}{2} \ln\left(1 +
\frac{b_{max}^2}{b_{min}^2} \right)
.. math::
b_{min} \equiv \sqrt{λ_{de Broglie}^2 + ρ_⟂^2}
.. math::
b_{max} \equiv \sqrt{λ_{Debye}^2 + a_i^2}
This method is valid for any plasma because it is impossible for
:math:`\ln{Λ} < 0` by this method, even if the coupling
parameter is high (such as for nonideal, dense, cold plasmas).
Note: This is the sixth method in Table 1 of
:cite:t:`gericke:2002`.
Examples
--------
>>> import astropy.units as u
>>> n_e = 1e19 * u.m**-3
>>> T = 1e6 * u.K
>>> Coulomb_logarithm(T, n_e, ('e-', 'p+'))
14.545527...
>>> Coulomb_logarithm(T, n_e, ('e-', 'p+'), V = 1e6 * u.m / u.s)
11.363478...
See Also
--------
~plasmapy.formulary.collisions.lengths.impact_parameter : Computes
:math:`b_{min}` and :math:`b_{max}`.
"""
# fetching impact min and max impact parameters
bmin, bmax = lengths.impact_parameter(
T=T, n_e=n_e, species=species, z_mean=z_mean, V=V, method=method
)
if method in (
"classical",
"ls",
"ls_min_interp",
"GMS-1",
"ls_full_interp",
"GMS-2",
):
ln_Lambda = np.log(bmax / bmin)
elif method in ("ls_clamp_mininterp", "GMS-3"):
ln_Lambda = np.log(bmax / bmin)
if np.any(ln_Lambda < 2):
if np.isscalar(ln_Lambda.value):
ln_Lambda = 2 * u.dimensionless_unscaled
else:
ln_Lambda[ln_Lambda < 2] = 2 * u.dimensionless_unscaled
elif method in (
"hls_min_interp",
"GMS-4",
"hls_max_interp",
"GMS-5",
"hls_full_interp",
"GMS-6",
):
ln_Lambda = 0.5 * np.log(1 + bmax**2 / bmin**2)
else:
raise ValueError(
'Unknown method. Choose from "classical", "ls_min_interp", '
'"ls_full_interp", "ls_clamp_mininterp", "hls_min_interp", '
'"hls_max_interp", "hls_full_interp", and their aliases. '
"Please refer to the documentation of this function for "
"more information."
)
ln_Lambda = ln_Lambda.to(u.dimensionless_unscaled).value
min_ln_Lambda = np.nanmin(ln_Lambda)
if min_ln_Lambda < 2 and method in (
"classical",
"ls",
"ls_min_interp",
"GMS-1",
"ls_full_interp",
"GMS-2",
):
warnings.warn(
f"The calculation of the Coulomb logarithm has found a value of "
f"min(ln Λ) = {min_ln_Lambda:.4f} which is likely to be inaccurate "
f"due to strong coupling effects, in particular because "
f"{method = } assumes weak coupling.",
CouplingWarning,
)
elif min_ln_Lambda < 4:
warnings.warn(
f"The calculation of the Coulomb logarithm has found a value of "
f"min(ln Λ) = {min_ln_Lambda:.4f}. Coulomb logarithms of ≲ 4 may "
f"have increased uncertainty due to strong coupling effects.",
CouplingWarning,
)
return ln_Lambda
[docs]
@validate_quantities(impact_param={"can_be_negative": False})
def Coulomb_cross_section(impact_param: u.Quantity[u.m]) -> u.Quantity[u.m**2]:
r"""
Cross-section for a large angle Coulomb collision.
Parameters
----------
impact_param : `~astropy.units.Quantity`
Impact parameter for the collision.
Returns
-------
sigma : `~astropy.units.Quantity`
The Coulomb collision cross section area.
Notes
-----
The `collisional cross-section
<https://en.wikipedia.org/w/index.php?title=Cross_section_(physics)&oldid=1037954726#Collision_among_gas_particles>`__
for a 90° Coulomb collision is obtained by
.. math::
σ = π (2 * ρ_⟂)^2
where :math:`ρ_⟂` is the distance of the closest approach for a 90°
Coulomb collision. This function is a generalization of that
calculation. Please note that it is not guaranteed to return the
correct results for small angle collisions.
Examples
--------
>>> Coulomb_cross_section(7e-10 * u.m)
<Quantity 6.157...e-18 m2>
>>> Coulomb_cross_section(0.5 * u.m)
<Quantity 3.141... m2>
"""
return np.pi * (2 * impact_param) ** 2