coupling_parameter

plasmapy.formulary.collisions.dimensionless.coupling_parameter(T: ~astropy.units.quantity.Quantity, n_e: ~astropy.units.quantity.Quantity, species, z_mean: float = nan, V: ~astropy.units.quantity.Quantity = <Quantity nan m / s>, method='classical') → Quantity[source]

Ratio of the Coulomb energy to the kinetic (usually thermal) energy.

Classical plasmas are weakly coupled (\(Γ ≪ 1\), where \(Γ\) is the coupling parameter). Dense plasmas tend to have significant to strong coupling (\(Γ ≥ 1\)). For more details, see the notes section below.

Parameters:

  • T (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 (Quantity) – The electron number density in units convertible to per cubic meter.

  • species (tuple) – A tuple containing string representations of the test particle (listed first) and the target particle (listed second).

  • z_mean (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 (Quantity, optional) – The relative velocity between particles. If not provided, thermal velocity is assumed: \(μ V^2 \sim 2 k_B T\) where \(μ\) is the reduced mass.

  • method (str, optional) – The method by which to compute the coupling parameter: either "classical" or "quantum". The default method is "classical". The Notes section of this docstring has more information about these two methods.

Returns:

coupling – The coupling parameter for a plasma.

Return type:

float or ndarray

Raises:

  • ValueError – If the mass or charge of either particle cannot be found, or any of the inputs contain incorrect values.

  • UnitConversionError – If the units on any of the inputs are incorrect.

  • TypeError – If any of n_e, T, or V is not a Quantity.

  • RelativityError – If the input velocity is same or greater than the speed of light.

Warns:

  • UnitsWarning – If units are not provided, SI units are assumed.

  • RelativityWarning – If the input velocity is greater than 5% of the speed of light.

Notes

The coupling parameter is given by

\[Γ = \frac{E_{Coulomb}}{E_{Kinetic}}\]

The Coulomb energy is given by

\[E_{Coulomb} = \frac{Z_1 Z_2 q_e^2}{4 π ε_0 r}\]

where \(r\) is the Wigner-Seitz radius, and 1 and 2 refer to particle species 1 and 2 between which we want to determine the coupling.

In the classical case the kinetic energy is the thermal energy:

\[E_{kinetic} = k_B T_e\]

The quantum case is more complex. The kinetic energy is dominated by the Fermi energy, modulated by a correction factor based on the ideal chemical potential. This is obtained more precisely by taking the thermal kinetic energy and dividing by the degeneracy parameter, modulated by the Fermi integral [Gericke et al., 2002]:

\[E_{kinetic} = 2 k_B T_e / χ f_{3/2} (μ_{ideal} / k_B T_e)\]

where \(χ\) is the degeneracy parameter, \(f_{3/2}\) is the Fermi integral, and \(μ_{ideal}\) is the ideal chemical potential.

The degeneracy parameter is given by

\[χ = n_e Λ_{de Broglie} ^ 3\]

where \(n_e\) is the electron density and \(Λ_{de Broglie}\) is the thermal de Broglie wavelength.

See equations 1.2, 1.3 and footnote 5 in Bonitz [1998] for details on the ideal chemical potential.

Examples

>>> import astropy.units as u
>>> n = 1e19 * u.m**-3
>>> T = 1e6 * u.K
>>> species = ("e", "p")
>>> coupling_parameter(T, n, species)
<Quantity 5.8033...e-05>
>>> coupling_parameter(T, n, species, V=1e6 * u.m / u.s)
<Quantity 5.8033...e-05>